Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UFE0022874/00001
## Material Information- Title:
- Essays in Applied Econometrics
- Creator:
- Zhou, Qiong
- Place of Publication:
- [Gainesville, Fla.]
- Publisher:
- University of Florida
- Publication Date:
- 2009
- Language:
- english
- Physical Description:
- 1 electronic resource
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Economics
- Committee Chair:
- Ai, Chunrong
- Committee Members:
- Hamilton, Jonathan H.
Terza, Joseph V. Shen, Wei - Graduation Date:
- 5/2/2009
## Subjects- Subjects / Keywords:
- Business executives ( jstor )
Business structures ( jstor ) Censorship ( jstor ) Chief executive officers ( jstor ) Consistent estimators ( jstor ) Corporate mergers ( jstor ) Estimators ( jstor ) Human capital ( jstor ) Maximum likelihood estimations ( jstor ) Target acquisitions ( jstor ) Economics -- Dissertations, Academic -- UF acquiring, acquisition, asymptotic, censored, ceo, consistency, data, distribution, estimation, firm, likelihood, maximum, merger, model, panel, simulation, tobit, turnover - Genre:
- Electronic Thesis or Dissertation
Economics thesis, Ph. D.
## Notes- Abstract:
- My research examines three separate studies of applied econometrics. In the first study, I empirically assess the impact of human capital acquisition from the target firm through a merger or an acquisition on post merger CEO turnover in the acquiring firm. Little is known about the effects of a merger or an acquisition on the acquiring firmâ€™s management team. The empirical evidence shows that merger is a way to acquire talented human capital, which will change both top management team and board structure of the acquiring firm, and thus result in leadership change in the acquiring firm. Using a sample of 236 mergers during 1996 to 2000 in the US, I find: (1) 46% of CEOs of acquiring firms are replaced within 5 years; 28% leave voluntarily, and 18% are forced to step down; (2) if the target firm's top executives are retained as top executives, the acquiring firm's CEO is more likely to leave; (3) if top executives of the target firm are retained as board directors in the acquiring firm, the acquiring firm's CEO is less likely to leave voluntarily, but no change occurs in the probability of being forced out. ( ,, )
- Abstract:
- Next, I investigate the finite sample performance of several estimators proposed for the panel data Tobit regression model with individual effects, including the Honoré estimator and the continuously updating GMM estimator. The continuously updating GMM estimator is based on more conditional moment restrictions than the Honoré estimator, and consequently is more efficient than the Honoré estimator for large samples. My simulation study shows that the continuously updating GMM estimator performs not better, but in most cases worse than the Honoré estimator for small samples. The reason for this finding is that the continuously updating GMM estimator is based on more moment restrictions that require discarding observations. In my design, about seventy percent of observations are discarded. The too few observations lead to an imprecise weighting matrix estimate, which in turn leads to an unreliable updating GMM estimator. This study calls for an alternative estimation method that does not rely on trimming.
- Abstract:
- In the final study, I propose a maximum likelihood estimator (MLE) for the panel data Tobit regression model with unknown individual effects. To overcome the problem occurred in chapter 2, my proposal is to use log likelihood density function instead of conditional moment restrictions in optimization problem. I suggest to approximate unknown density function of individual effects with a sieve estimator and to estimate finite dimensional unknown parameters and infinite dimensional sieve estimator jointly by applying the method of maximum likelihood estimation. Under some sufficient conditions, I show that (1) the sieve estimator of unknown density function for individual effects is consistent under certain metric; (2) the MLE estimators of the finite dimensional parameters are consistent and asymptotically normally distributed; (3) the estimator for the asymptotic covariance of the parameter is consistent.
- Statement of Responsibility:
- by Qiong Zhou
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Qiong Zhou. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 5/31/2011
- Resource Identifier:
- 780092460 ( OCLC )
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PAGE 1 1 ESSAYS IN APPLIED ECONOM E TRICS By QIONG ZHOU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 PAGE 2 2 2009 Qiong Zhou PAGE 3 3 To my parents, Min Zhe ng and Yuanguang Zhou PAGE 4 4 ACKNOWLEDGMENTS The successful completion of this dissertation was not possible without support of several individuals. E ach of these individual s has provided endless support and invaluable suggestions. F irst and foremost, I am deeply indebted to my chair, Dr. Chunrong Ai who opened the door of graduate study in the United States for me. Without his incredible support and guida nce, I could not have completed thi s work I cannot thank him enough for his help through this process. I thank Dr. Wei Shen for bringing me into the new field, strategic management and for his wonderful guidance on my future career. I thank Dr. Jonathan Hamilton for providing useful suggestions and support throughout my graduate work I also thank Dr. Joseph Terza for great suggestions I thank all my friends, here and in China, whose support was crucial during my life in the United States Finally, I tha nk my family my parents, grandparents, a unties and uncles. Their unconditional love and support guided me all my life. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................................... 4 LIST OF TABLES ................................................................................................................................ 7 LIST OF FIGURES .............................................................................................................................. 8 ABSTRACT .......................................................................................................................................... 9 CHAPTER 1. HUMAN CAPITAL ACQUISITION AND POST MERGER TURNOVER OF ACQUIRING FIRMS CEO ...................................................................................................... 11 Introduction ................................................................................................................................. 11 Human Capital Acquisition ........................................................................................................ 14 Human Capital Acquisition a s a Top Team Officer .......................................................... 17 Human Capital Acquisition a s a Board Member ............................................................... 18 Data and Empirical Design ......................................................................................................... 19 Definition of CEO Turnover ............................................................................................... 20 Human Capital Acquisition ................................................................................................. 22 Empirical Design and Control Variables ........................................................................... 24 CEO s age ..................................................................................................................... 25 Firm performance ......................................................................................................... 27 Corporate governance variables .................................................................................. 28 Empirical Results ........................................................................................................................ 33 Logit Estimates of CEO Turnover ...................................................................................... 33 Multinomial Logit Estimates of CEO Turnover Type ...................................................... 36 Conclusion ................................................................................................................................... 41 2. ESTIMATION OF CENSORED REGRESSION MODEL: A SIMULATION STUDY ...... 52 Introduction ................................................................................................................................. 52 Model and Estimators ................................................................................................................. 53 Honor Estimation ............................................................................................................... 57 GMM Estimation ................................................................................................................. 58 Empirical Likelihood Estimation ........................................................................................ 59 Monte Carlo Experiments ........................................................................................................... 61 Design 1 ................................................................................................................................ 61 Design 2 ................................................................................................................................ 62 Design 3 ................................................................................................................................ 63 Design 4 ................................................................................................................................ 63 Conclusion ................................................................................................................................... 67 3. MAXIMUM LIKELIHOOD ESTIMATION OF Panel Data TOBIT MODEL ..................... 79 PAGE 6 6 Introduction ................................................................................................................................. 79 MLE Estimator ............................................................................................................................ 81 Consistency .................................................................................................................................. 86 Asymptotic Distribution ............................................................................................................. 92 Covariance Estimator .................................................................................................................. 96 Conclusion ................................................................................................................................... 96 APPENDIX: PROOF FOR CHAPTER 3 ......................................................................................... 99 LIST OF REFERENCES ................................................................................................................. 106 BIOGRAPHICAL SKETCH ........................................................................................................... 113 PAGE 7 7 LIST OF TABLES Table page 1 1 Sample distribution and frequency of the CEO turnover .................................................. 43 1 2 Descriptive statistics for human capital acquisition ........................................................... 44 1 3 Variable definitions .............................................................................................................. 45 1 4 Explanatory variable descriptive statistics for the total sample ........................................ 46 1 5 Correlations .......................................................................................................................... 47 1 6 Results of Logit Regression Model ..................................................................................... 48 1 7 Results of multinomial logistic regression model .............................................................. 49 2 1 Censored observations a nd discarded observations ........................................................... 71 2 2 Monte Carlo study for Honor and the updating GMM estimator beta1 in design 1 ...... 72 2 3 Monte Carlo stud y for Honor and the updating GMM estimator beta2 in design 1 ...... 73 2 4 Monte Carlo study for Honor and the updating GMM estimator beta1 in design 2 ...... 74 2 5 Monte Carlo study for Honor and the updating GMM estimator beta2 in design 2 ...... 75 2 6 Monte Carlo study for Honor and the updat ing GMM estimator beta1 in design 3 ...... 76 2 7 Monte Carlo study for Honor and the updating GMM estimator beta2 in design 3 ...... 77 2 8 A 200 Observations Sample ................................................................................................ 78 PAGE 8 8 LIST OF FIGURES Figure pag e 2 1 Discarding observations when 00ix ............................................................................ 69 2 2 Discarding observations when 00ix ............................................................................ 70 PAGE 9 9 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ESSAYS IN APPLIED ECONOM E TRICS By Qiong Zhou May 2009 Chair: Chu nrong Ai Major: Economics My research examine s three separate studies o f applied econometrics. In the first study, I empirically assess the impact of human capital acquisition from the target firm through a merger or an acquisition on post merger CEO turnover in the acquiring firm. Little is known about the effects of a merger or an acquisition on the acquiring firms management team. The empirical evidence shows that merger is a way to acquire talented human capital, which will change both top manage ment team and board structure of the acquiring firm, and thus result in leadership change in the acquiring firm. Using a sample of 236 mergers during 1996 to 2000 in the US, I find: (1) 46% of CEOs of acquiring firms are rep lace d within 5 years ; 28% leave voluntar ily and 18% are forced to step down; (2 ) if the target firm's top executives are retained as top executives, the acquiring firm's CEO is more likely to leave; (3) if top executives of the target firm are retained as board directors in the acquiri ng firm the acquiring firm's CEO is less likely to leave voluntarily, but no change occurs in the probability of being forced out. Next, I in vestigate the fi nite sample performance of several estimators proposed for the panel data Tobit regression model with individual eff ects, including the Honor est imator and the continuously updating GMM estimator The continuously updating GMM est imator is based on more conditional moment restrictions than the Honor estimato r, and consequently is more PAGE 10 10 efficient than the Hon or estimator for large samples. My simulation study shows that the conti nuously updating GMM estimator per forms not better, but in most cases worse than the Honor estimator for small samples. The reason for this fi nding is that the continuously updating GMM estimator is based on more moment restrictions that require discarding observations. I n my design, about s even ty percent of observations are discarded. The too few observations lead to an imprecise weighting matrix esti mate, which in turn lead s to an unreliable updating GMM estimator. This study calls for an alternative estimation method that does not rely on trimming. In the final study I propose a maximum likelihood estimat or (ML E) for the panel data Tobit regression model with unknown individual eff ects To overcome the problem occurred in chapter 2, my proposal is to use log likelihood density function instead of conditional moment restrictions in optimization problem. I suggest to approximate unknown density function of individual effects with a si eve estimator and to estimate finite dimensional unknown parameters and infinite dimens ional sieve estimator jointly by applying the method of maximum lik elihood estimation Under some sufficient conditions, I show that (1) the sieve estimator of unknown density function for individual effects is consistent under certain metric ; (2) the MLE estimator s of the finite dimensional parameter s are consistent and asymptotically normally distributed; (3) the estimator for th e asymptotic covariance of the p arameter is con sistent. PAGE 11 11 CHAPTER 1 HUMAN CAPITAL ACQUISITION AND POST MERGER TURNOVER OF ACQUIRING FIRM S CEO Introduction The past decades have witnessed a great wave of merger and acquisition activities. In the US alone, over 200,000 deals occurred between 1963 and 2007. Much of the existing literature focuses on understanding the organizational changes within the acqui ring firm after a merger1, parti c u l arly leadership change. It is important to grasp the possible implications that the merger will have on the top management team. In the literature on post merger managerial turnover, a large body of studies examine post merger t urnover of the tar get firm's top executives Three theories have been put forth to explain why some top executives from the target firm leave after the merger Market discipline theory, for one, argues that the market for corporate control plays an important disciplinary ro le. Ineffective managers of target firms who performed poorly before the merger are likely to be replaced aft er the merger ( e.g., Walsh and Ellwood, 1991; Martin and McConnel, 1991; Hadlock, Houston, and Ryngaert, 1999; Harford, 2003). This theory is suppo rted by empirical evidence ( e.g., Coughlan and Schmidt, 1985; Warner, Watts and Wruck, 1988; Weisbach, 1988; Gibbons and Murphy, 1990; Blackwell, Brickley and Weisbach, 1994). Relative standing theory or local social status theory, for another, suggests that, if the acquired executives feel inferior, are stripped of status, or locked in a struggle with the acquirers in the merged entity, they will tend to leave post merger ( e.g. Hambrick and Cannella, 1993; Lubatkin, Schweiger, and Yaakov, 1999; Cannella and Hambrick, 1993). 1 For convenience, I will refer to all of the mergers and acquisitions studied as mergers. Below, I discuss the definition of the acquiring firm and the target firm (which applies to all mergers in the data). PAGE 12 12 A third argument comes from human capital theory. I f the cost of human capital investment that the acquired executives have to spe nd post merger is larger t han the potent ial future returns, the executives from the target firm are likely to leave (e.g., Buchholtz, Ribbens and Houle, 2003). While it is important to study the fate of the target firms CEO post merger it is equally important to observe what happens to the acquiring firms CEO post merger Little research has investigated the impact of a merger on the acquiring firms CEO; the only exception is Lehn and Zhao (2006). They examine the relation between cumula tive abnormal returns in the stock market following acquisition announcement and subsequent CEO turnover They find that if the acquiring firm's CEO makes an acquisition that creates shareholder value he is expected to be rewar ded with extended tenure. I n contrast, if the acquiring firm's CEO makes an acquisition that reduces shareholder value, he or she will be punished and replaced. Their argument follows the traditional market discipline perspective of CEO turnover. To broaden understanding of the impac t of a merger on post merger turnover of the acquiring firm's CEO, in this paper I investigate an alternative perspective on how human capital acquisition from the target firm through a merger affect s CEO turnover in the acquiring firm post merger Bell A tlantics merger with NYNEX in 1996 supports this view. After the merger, Ivan G. Seidenberg, chairman and chief executive officer of NYNEX, had been retained as vice chairman, president and chief operating officer. Frederic V. Salerno, who was vice chairm an of NYNEX, became the new chief financial officer and executive vice president. About one and half years after the closing of the merger, according to the terms of the agreement, Mr. Seidenberg became chief executive officer of the new company and chairm an upon Raymond PAGE 13 13 W. Smith's retirement, who was chairman and chief executive officer of Bell Atlantic at the time of the merger announcement One of the motivations for a merger is to acquire talented and skillful top executives from the target firm, especi ally to obtain potential successors for the current CEO The to p executives who come from the target firm can provide valuable experti se for firm strategy and supply complementary knowledge that is not available within the acquiring firm to help the newly combined firm to survive. In addition, human capital acquisition through merger changes the distribution of power and control within the acquir ing firm. With a better potential CEO candidate the incumbent CEO is easier to replace if his p erformance drops. Moreover, the acquiring firm would rely less on the incumbent CEO. The new "upper echelon" members could also cause a power struggle within the top management team of the newly combined firm. If the CEO of the acquiring firm is successfully challenged by the acquired top executives and loses control, or fails in a power struggle with the acquired manag ers from the target firm, he is likely to depart. To test whether this human capital acquisition theory can explain post merger turnover of the acquiring firm's CEO, I collected a sample of merger s and acquisitions. Of the 236 mergers that occurred from 1996 to 2000, there are 109 or 46% of cases where the CEO of the acquiring firm was replaced within five years of the merger announce ment. Of thos e 109 cases, 42 were forced to depart, and 67 were replaced voluntarily. I employ both logit and multinomial logit regression model s to exami ne the relationship between post merger turnover of the acquiring firm's CEO and human capital acquisition from the target firm through merger and acquisition activities. Bo th regression results show that if the target firm's top executive is retained as a top executive in the merged entity after the merger PAGE 14 14 the acquiring firm's CEO is more likely to leave, either volu ntarily or involuntarily. If the target firm's top executives are retained as board directors in the merg ed firm, the acquiring firm's CEO is less likely to leave voluntarily, but no change occurs in the likelihood of being forced out No significant assoc iation exists between an acquiring firm's board characteristics and the likelihood that the acquiring firms CEO will be replaced Finally, the more ownership concentration in the acquiring firm the less the effect of director acquisition is on post merge r CEO turnover. My study contributes to the literature by further studying the role that a merger plays in post merger CEO turnover in the acquiring firm. Based on this study, another theory about human capital acquisition could be added alongside the mar ket discipline theory to explain CEO turnover in the acquiring firm post merger Also, the current study suggests that Shleifer and Vishny's (2003) theory of "stock market driven acquisitions" is incomplete. The merger does not only result in market takeover, but also fulfills the acquiring firms desire for human capital talent. The paper is organized as f ollows: Section 2 discusses the hypotheses, Section 3 describes the sample and empirical d esign use s in the paper, Section 4 discusses the empirical resu lts, and Section 5 concludes the paper. Human Capital Acquisition The existing studies find that a firm's good performance is a reflection of a good and efficient top management team, including the CEO, president, chairman of the board, vice president s C FO COO, and other "upper echelon" executives. The top executiv es are unique organizational resources (e.g., Daily, Certo, and Dalton, 2000). This unique human capital can have major impacts on organizational actions and performances (e.g., Thompson, 1967). When these resources are aligned with the organizational goal of the acquiring firm, they are much PAGE 15 15 more productive than general labor and are more likely to produce a competitive advantage for the firm (Hitt, Bierman, Shimize, & Kochhar, 2001). Thus to gain a competitive edge, it is critical to have a talented top management team. However, talented top executives are few a nd in high demand ; good heirs apparent are especially hard to obtain. One way to acquire highly talented and skillful top executi ves is through merger (e.g., Parons and Baumgartner, 1970; and Pitts, 1976). Acquiring successful and skillful top managers from the target firm to replace the ineffective incumbent top managers or CEO in the acquiring firm has several advantages. First a n acquired top manager can be a good CEO heir apparent for the acquiring firm, especially when the incumbent CEO of the acquiring firm is close to retirement Many firms struggle to smooth the power transition process by choosing a n heir apparent well in a dvance of the actual CEO turnover (Wall Street Journal, 1997). Merger provides a way to find a good heir apparent since he can work together with the i ncumbent CEO as the CEO passes power and control Second the acquired executives will provide valuable e xpertise for firm strategy and supply complementary knowledge that is not available within the acquiring firm. Moreover, merger and acquisition will not only change the acquiring firm's "upper echelon by acquiring talented executives from the target firm, b ut also elsewhere in the organization. Such changes in the organization frequently lead to critical problems, difficulties, uncertainties and contingencies. The top management team self -perceived capacity or incapacity for dealing with the critical issue is important for the newly combined firm to survive. Top executives from the target firm who are capable of coping with the new organization's environment could be selected to enter the "upper echelon" and capture power and controls within the firm. Third such human capital acquisition also brings the incentive to the incumbent top executives to improve the firm's PAGE 16 16 performance. Hence, human capital acquisition can influence post merger turnover of the acquiring firm's CEO through the internal governance me chanism. The disadvantage of acquiring top executives through merger is that the acquired ex ecutives take power and control from incumbent top managers and the CEO hence exacerbating the power contest between top managers One outcome of the redi stributio n of power and control within the firm deriving from human capital acquisition through merger is CEO turnover in the acquiring firm post merger Many studies find that the competition among top executives plays an important role in CEO dismissal and succes sion decisions (Boeker, 1992; Cannella & Lubatkin, 1993; Ocasio, 1994; Cannella & Shen, 2001; Shen & Cannella, 2002). Acquired top executives confront significant challenges upon taking office af ter entering the "upper echelon" of the acquiring firm. To keep the position, the acquired executives might want to obtain more power within the "upper echelon", and have more desire for career advancement and demand better performance. In addition, since acquired top executives are new to the incumbent top executives, they are more likely to have different interests and strategies than the incumbent CEO. Thus, newly joined top executives from the target firm are more likely to challenge the CEO and worsen the power competition. Although such contests a re not easily observable, Shen and Cannel la (2002) argued that the power contests among top executives will affect the process of the CEO dismissal. When acquired top executives successfully challenge the CEO, the CEO will be dismissed (Preffer, 1981; Sonnenfeld, 1988). Furthermore, when acquired top executives are locked in a struggle with the acquirer, the power contest and interest conflict can harm the firm's performance. A f irm's poor performance results not only because of an ineffecti ve CEO, but also due to the power t ournament within top PAGE 17 17 management, which normally is the reason for CEO turnover. Therefore, human capital acquisition will affect post merger turnover of the acquiring firm's CEO. Insofar as both advantages and disadvantag es of human capital acquisition will influence post merger turnover of the acquiring firm's CEO, it is an ideal topic to examine how human capital acquisition decision s through merger affect the acquiring firm's CEO replacement. Top executive acquisition f rom the target firm provides an opportunity to examine the relation between human capital acquisitions through merger and the acquiring firm's CEO turnover after merger When acquiring firms decide to retain valuable top executives from the target firm, ac quired top executives may get positions in the acquiring firms' top management team after merger where they may control the firm's strategy, or they may enter the board of the acquiring firm where they have power to monitor and advise the CEO. Both types of human capital acquisition from the target firm are expected to have some influence on the acquiring firm's incumbent CEO turnover. H uman Capital A cquisition a s a Top Team O fficer If a top executive of the target firm is retained as a top team management officer in the acquiring firm after merger it means that the acq uiring firm believes that he is a good manager or potential successor T he firm with a CEO who is going to retire within a few years would like to pass power and control of the firm to a potential successor With high status and more power within the "upper echelon" of the acquiring firm, the prospective CEO heir would be able to hold the highest position in the firm earlier by subverting the power competition within the top management tea m, while it is hard for the current CEO to maintain independent control of the firm when there are new top executives acquired from the target firm. On the other hand, acquired to p executives from the target firm are believed to be more capable of co ping w ith the newly combined firm compared to the current members of the top management team. Therefore, PAGE 18 18 power competition is going to be escalate d by the new competent "upper echelon." With more power contests, the firm's strategy may not be effici ent and compe titive. F or instance, to compete with ne w members of the "upper echelon," the current CEO may be involved in high risk projects and make wrong decisions, thus the firm's performance will decline, which could lead hi m to be replaced. Moreover, compared to t he firm without potential successors, when a firm has good candidates for CEO succession, the board would be more likely to replace the CEO who does not perform well As a result, I expect t hat human capital acquisition of top team officers has a positive effect on post merger turnover of the acquiring firm's CEO. Hypothesis1: If the target firm's top executive has retained top executives in the newly combined firm after the merger, the acquiring firm's CEO is more likely to leave, either voluntarily or inv oluntarily. H uman Capital A cquisition a s a B oa rd M ember For most of the large merger s, top executives of the target firm, such as the CEO and president who are also shareholders before merger may become directors on the acquiring firm's board after the de al is completed, especially when the deal is paid for with stock. Therefore, given the context of merger it is interesting to study the effect of such human capital acquisition on post merger turnover of the acquiring firm's CEO through the internal gover nance mechanism. The CEO who is going to retire within a short time is normally older and less aggressive, thus he would like to make a friendly merger and provide a better deal to the target firm's top managers as compared to an ambitious CEO. Thus, top executives of the target firm who have positions on the board of the acquiring firm after the merger could have a good relationship with the incumbent CEO in the acquiring firm. The target firm's top executives would appreciate for being provid ed a good deal and a position on the board of the acquiring firm after the merger PAGE 19 19 was effected. The current CEO could also benefit from the good relationship since those new directors who are from the top management team of the target firm could provide useful advice about managing the acquired firm and support them in the board. Thus, I expect that if acquiring firms make director acquisition through merger the likelihood of the acquiring firms CEO being replaced decreases. Hyp othesis2: If the acquiring firm acquired the target firm's top executives and they are listed as board directors in the newly combined firm after the merger, the acquiring firm's CEO is less likely to leave. Data and Empirical Design The sample of merger s is obtained from Thomson Financial Securities Data Corporate (SDC) MERGER database, COMPUSTAT, and Disclosure's Compact D SEC database2. I begin withdrawing the initial sample from SDC based on the following criteria: (1) the merger or acquisition is annou nced between January 1, 1996 and December 31, 20003; (2) The transaction occurs in the US ; (3) the form of the deal is merger or acquisition; (4) the status of the deal is "completed"; (5) both the acquiring and target firm are publicly traded4; (6) the bu yer's net sales in the last twelve months is greater than 100 million dollars. These screens yield a candidate sample of 1480 merger s. In order to ensure that the deal represents "large" investments by acquiring firms, I require that the size of the target firm is at least 30% of the size of the acquiring firm, measured by net assets. The acquiring firm would be more likely to acquire human capital from the target firm 2 I accessed all databases through the UFL business library. 3 I ended the period of the sample at 2000, so that I could observe the CEO turnover information within five years following merger announcement. 4 For deals as form of merger SDC merger and acquisition dataset has already distinguished between the acquirer and target. So I followed their criterion of which firm is acquirer and which firm is target for merger transaction. PAGE 20 20 when the deal is large enough, or as a form of merger especially for the human capital a cquisition as directors. 408 deals satisfied this requirement. Information about CEO turnover of the acquiring firm and top management team retention of the target firm are also required to be available to each acquiring and target firm on the deal's announcement year through five years after the announcement date. The main sources of the information about CEO and top management team are company annual reports and proxy statements in the SEC Edgar database. The SEC Edgar database is also the source of infor mation about firms' governance structures. After searching company annual reports, proxy statements, LexisNexis and news wires, 104 merger s are excluded since CEO turnover information couldn't be identified or because the governance data is incomplete. As a result, this filter reduces the sample to 304 merger s. Moreover, both firms have to be listed on the COMPUSTAT financial statement data for each acquiring and target firm could be collected The final sample for regression analysis includes 236 mergers m ade during the sample period. Defi nition of CEO Turnover Turnover is classified into two types. I define "CEO turnover" to include CEO replacement within five years after merger announcement that reported as retired, or replaced but still served on the board, resigned or terminated. For five CEO replacements, which are reported as deceased, I classify the observations as one censored. Therefore, the "no CEO turnover" group includes those transactions where the CEO who announced the merger wa s still the CEO of the acquiring firm after five years since the announcement, or there wa s a CEO replacement due to death or poor health. The "CEO turnove r" group includes the other transactio ns where the CEO who announced the merger wa s replaced within five years after the merger announcement. PAGE 21 21 The types of CEO turnover are identified by voluntary turnover and forced turnover. "F orced CEO turnover under the subsample "CEO turnover," is define d a s a non routine CEO replacement, i.e., the CEO wa s resigned, or terminated within five years of merger For the other CEO turn over, if the departing CEO wa s reported as retired within five years of the merger or if t he news reports that the CEO would s till serve on the board as a non -executive chair or vice chair, then the CEO turnover is classified as a "voluntary turnover." To identify the types of CEO turnover, the acquiring firm's proxy statements are examined for the announcement year and five yea rs after the announcement. Of the 408 deals, CEO turnover type for 104 transactions could not be identified. For 29 deals information about the CEO at the announcement year could not be found 44 deals have incomplete information about CEO turnover within five years in the dataset, and 31 were acquired by other firms within five years5. Therefore, these 104 deals were excluded from the sample. A dummy variable is created for CEO turnover that takes value of one if the acquiring firm's CEO is replaced within five years of the merger 's announcement, and zero otherwise. Also, a categorical variable is created for the turnover type of the acquiring firm's CEO that takes the value of one if the acquiring firm's CEO is replaced voluntarily within five years of the announcement, and two if the acquiring firm's CEO is fired or forced to step down, and zero if there is no turnover. Table 1 1 presents descriptive statistics for post merger turnover of the acquiring firm's CEO. Panel A of Table 1 1 reports the frequenc y of "CEO turnover" versus "no CEO turnover" for the sample. Of the 236 merger s in the sample, 127 are not replaced after their respective 5 Since this paper examines the relation between human capital acquisit ion and CEO turnover, the effect of human capital acquisition is through internal governance. In addition, this paper wants to examine the different effect of human capital acquisition on the type of CEO turnover. Thus, I exclude those merger s in which CEOs are replaced PAGE 22 22 acquisitions within five years, 109 are replaced within five years. Within the subsample of 109 observations subject to CEO turnover, 67 are replaced voluntarily within five years, and the remaining 42 are forced to step down within five years after their respective merger This distribution indicates that close to half of the CEOs in the full sample are subjected to rep lacement within five years of the merger s, and among them, about sixty percent of CEOs are replaced voluntarily. Panel B reports the frequency of CEO turnover across different merger announcement year for the full sample of 236 merger s, and for four subsamples. For merger s announced in 1996 and 1999, the probability of post merger CEO turnover in the acquiring firm is higher compared to the mergers announced in 1997, 1998 and 2000, where more than half of CEOs are subjected to replacement within five years. Of mergers announced in 1996, over forty percent the highest of the five years of CEOs from the acquiring firm voluntarily stepped down. And of mergers announce d in 1999, about thirty percent t he highest among the five years -of CEOs from the acquiring firm were forced to step down. Human Capital A cquisition To examine the effect of human capital acquisition from the target firm on turnover of the acquiring firm's CEO post merger I hand -coded two variables not previously studied: "officer acquis ition" and "director acquisition." I first collected the names of top management officers of the target firm reported in the proxy statements from the SEC dataset in the merger announcement year, and then collected names of both top management officers and board directors of the acquiring firm after the merger was effected. Then I checked whether the top by ext ernal control market, i.e., takeover or bankruptcy. That is, I collect data that CEOs are only replaced by internal governance. PAGE 23 23 manager from the target firm got a position in the acquiring firm's top management team or board after the merger was completed. "Officer a cquisition is defined as a dummy variable6 which takes the value of one if one or more top managers of the target firm have been acquired as top managers in the acquiring firm after the merger was effected, and takes the value of zero otherwise. Similarly, "director ac quisition" is defined as a dummy variable7 which takes the value of one if one or more top managers of the target firm have been acquired as board directors in the acquiring firm after the merger was completed, and takes the value of zero otherwise. Table 1 2 reports the distribution of the human capital acquisition associated with merger for the entire sample and four subsamples. It lists the mean, median, and standard deviation values for both officer acquisition and director acquisition. On average, the probability that the acquiring firm would like to acquire top executive s from the target firm as new top managers in the acquiring firm is 0.28, and 0.35 for director acquisition. The probability of the acquiring firm with post merger CEO turnover would li ke to acquire human capital from the target firm is larger than the acquiring firm without CEO turnover, for both top officer acquisition and director acquisition. The probability of acquiring human capital from the target firm as top managers in the acqui ring firm is the highest for acquiring firms with forced CEO turnover (0.40), compared to acquiring firms with voluntary CEO turnover (0.28), firms with CEO turnover regardless of the turnover type (0.33), and firms without CEO turnover (0.24). 6 First, I collected the number of top managers in the target firm who have retained positions in the acquiring firm and were re ported as top management executives. However, in the regression model, I use a dummy variable rather than the number variable, because the existence of human capital acquisition as top officers have effect on CEO subsequent turnover, but the size of such human capital acquisition maybe not important in the model. The effect of human capital acquisition on CEO turnover may not vary over the size of human capital acquisition. 7 Similar to officer acquisition, I collected the number of top managers in the tar get firm who become directors on the merged entity's board after the merger was effected, and then scaled it by the acquiring firm's board size to PAGE 24 24 For directo r acquisition, namely, acquiring firms acquired top executives from the target firm to be new directors in the acquiring firm post merger ; the probabilities of such human capital acquisition are similar for three subsamples with CEO turnover. For instance, it is 0.40 for acquiring firms with forced CEO turnover, 0.38 for acquiring firms with voluntary CEO turnover, and 0.39 for acquiring firms with CEO turnover regardless of the turnover type. However, the probability of director acquisition for acquiring firms without CEO turnover is smaller (0.30). Empirical D esign and Control Variables To examine the relation between human capital acquisition from the target firm and the probability of the post merger turnover of the acquiring firm's CEO after merger wi thout specifying the turnover type, I estimate the logit model exp() (CEO turnover) 1exp() x prob x where the variable "CEO turnover" is defined as one if the CEO replaced within five years of the merger announcement, and zero if th k 1 vector of estimated coefficients for CEO turnover; x is a k 1 vector of explanatory variables which may influence the probability of CEO turnover according to empirical research on CEO turnover and merger issues, including human capital acquisition variab les, the acquiring firm's CEO's characteristics, the acquiring and the target firm's performance, and the acquiring firm's corporate governance. I also include four year dummies for the merger announcement years of 1997-2000 to account for aggregate change s over time8. Table 1 3 lists the definitions of all variables used in the model. eliminate the board size effect. Similar results are obtained if I use a scaled number of director acquisiti on instead of a dummy variable. Thus I use d a dummy variable in this paper. 8 1996 data is the base year indicated by all year dummies=0. PAGE 25 25 Table 1 4 provides descriptive statistics of relevant variables for the full sample, including mean, median, standard deviation, minimum and maximum values. Table 1 -5 lists t he correlations among all variables, including two different dependent variables, the post merger turnover of acquiring firm's CEO, and the turnover types of the acquiring firm's CEO within 5 years after the merger announcement. The correlations, shows tha t the turnover of the acquiring firm's CEO after merger was positively associated with human capital acquisition but not strong, for both officer and director acquisitions. The turnover type of the acquiring firm's CEO was positively associated with both o fficer and director acquisitions, but significant for top officer acquisition. In addition, the control variables, CEO age and ownership concentration, was significantly associated with both CEO turnover and CEO turnover type, as might be expected. And pos t -ROE of the acquiring firm was only significantly associated with CEO turnover type. There is no strong correlation for other control variables. The magnitudes of the correlations do not suggest that multicollinearity is an issue. CEO s a ge Buchholtz, Ribbens, and Houle (2003) find a significant relation between CEO age and CEO turnover, i.e. the probability of CEO departure will decrease with CEO age until a CEO reaches his/her middle age, and then the probability of CEO turnover will increase. After a merger CEO s needs to adjust to a new cast of characters require new investments to build new human capital. At an earlier age, CE Os might lack enough experience to handle the new challenge s that arise from merger activities. And, the time for younger CE Os to build enough human capital is too long a wait for the acquiring firm. Thus, the probability of forced turnover may be higher for CEOs who are relatively young The younger a CEO is, the less important the financial a nd career security is to him For younger CEOs, it is less painful to leave a position PAGE 26 26 and a company. Since younger CEOs can relocate relatively easily, the probability of voluntary CEO turnover may be higher for them. As a CEO grows older, past the middle age and approa ching the retiremen t age, he is less likely to make the new investment since fewer productive years of work are left. The acquiring firm would like to rely more on the new top executives, and decrease its dependence on the older CEO, especially for managing the new part of t he combined corporation -the acquired firm. In addition, merger activity, such a big change within the firm, would inevitably bring some risks. Older CEOs are expected to have less confidence to handle risk since they are less willing to build new human capital to deal with new risks. Thus, firms would like to diffuse their dependence on the current older CEO, and move some dependence to new top executives. Murphy and Zimmerman (1993), and Goyal and Park (2002) report a significant positive relation between CEO age and CEO turnover. Thus, I expect that an older CEO who passes middle age is more likely to retire. On the other hand, older CEOs are closer to their retirement age. With an older CEO, the acquiring firm has more pressure to look for a CEO heir a nd move power to the heir. Therefore, I expect the probability of voluntary turnover for the older CEO is higher as well. Therefore, the probability of both voluntary and forced CEO turnove r in the acquiring firm after a merger will decrease with age until a CEO reaches middle age, at which point the rate increases. I include both CEO age and CEO age squared at the year of the merger announcement in the analysis to model a curvilinear effect for CEO age. As seen in Table 1 4, the mean age of CEOs is 53.86 a nd the median is 54 for the full sample. The age of CEOs with turnover is significantly higher than those without turnover, especially for CEOs with voluntary turnover. The mean age of CEOs without turnover is 51.61, PAGE 27 27 while the mean age of CEOs with turnove r is 56.47, 58.39 for voluntary turnover, and 53.4 for forced CEO turnover. Similar results hold for the median age of CEOs. Firm performance Pre -merger performance of the acquiring firm : The literature states that a firm's performance commonly implies the CEO's ability Weisba ch (1988), Murphy and Zimmerman (19 93) find that the likelihood of CEO turnover is significantly higher when a firm's performance is lower Post merger turnover of th e acquiring firm's CEO after a merger might be due to the acquiring firm's poor performance before the corresponding merger I expect that the probabilities of both forced and voluntary post merger turnover of the acquiring firm's CEO are inversely related to the acquiring firm's pre -merger pe rformance. I include a measure of the acquiring firm's performance before the merger as an explanatory variable in the model. I calculated industry adjusted yearly return on common equity (pre ROE)9for one fiscal year prior to the merger announcement in th e acquiring firm10. Post merger performance of the acquiring firm : I also include a measure of the acquiring firm's performance after the corresponding merger as a control variable to capture the effect of the acquiring firm's post merger performance on post merger turnover of the acquiring firm's CEO. The turnover of the acquiring firm's CEO after merger might be due to the firm's poor performance after the corresponding merger event. I expect that the probability of both forced and voluntary CEO turnove r in the acquiring firm to be higher for those acquiring firms 9 Following Lehn and Zhao (2006), I calculate the industry adjusted accounting performance measures by subtracting th e industry median values from firm's corresponding measures' value. The industry median is the median value of the industry portfolio formed by matching the three digit SIC code from COMPUSTAT. 10 Return on assets (pre ROA), and return on average equity (pr e ROAE) are also available in my dataset. They all give similar results as pre ROE, thus I don't list them in the paper. PAGE 28 28 who have poor performance after merger Similarly, industry adjusted return on common equity (pre ROE) for the acquiring firm in the year right after the merger announcement are included. Pre -m erger performance of the target firm : Because my main interest is the effect of human capital acquisition from the target firm on post merger turnover of the acquiring firm's CEO, I also include a measure of the target firm's performance before the merger It is generally believed that a successfully performing firm usually has a competent and effectual top management team. T herefore, if a target firm has great performance before merger the acquiring firm would be like ly to obtain not only the assets of the target firm, but also its good top executives to replace the current top executives who are ineffectual or going to be retired soon in the acquiring firm. The acquired top executives may retain some feeling of centrali ty and importance, and they would get important positions in the combined enterprise's top management team post merger One might predict that when target firms have great performance, human capital acquisition through those target firms would be more skil lful and powerful which may induce a higher rate of post merger CEO departure in the acquiring firm. To control the effect of pre -merger performance of the target firm on post merger turnover of the ac quiring firm's CEO, I calculate a measure for the indus try adjusted return on common equity (pre -ROE of target) of the target firm in the year before the merger announcement. Table 1 4 reports the mean, median, and standard deviation values for the both acquiring and target firm's performance measures before a nd after the merger Corporate governance v ariables Because the effect of human capital acquisition on post merger turnover of the acquiring firm's CEO by internal governance mechanism, I examine whether the relation between human capital acquisition and p ost merger CEO turnover is related to the characteristics of corporate governance. Specifically, I examine the rol e the acquiring firm's board characteristics and PAGE 29 29 ownership structure play in the process. All governance data are taken from the acquiring fir m's proxy statement that is closest in time to the announcement of the corresponding merger from the SEC database. Board characteristics : Board size, board independence, and leadership are used to measure a firm's board structure. Board size is defined as the number of directors reported on the board. Board dependence is calculated as the percentage of inside directors on the board during the year of the merger announcement. Inside directors are defined as the board members who are employees, former employe es, employee' relatives, attorneys, or accountants. Leadership is a dummy variable to control leadership structure, which takes the value of one if the CEO of the acquiring firm also serves as the chairman on the board, and zero otherwise (e.g. Lehn & Zhao, 2006). Board size Yermack (1996) and Jensen (1993) find a significant inverse relation between board size and firm's performance. Small boards are more effective monitors so that can help to improve the CEO's performance. Thus, I expect that the probability of the post merger CEO turnover in acquiring firms with smaller boards should be higher than those with larger boards. Board in dependence The literature argues that dependent d irectors would decrease the board's monitoring function. CEO turnover is more sensitive to firm's performance when the board is more independent (Weisbach, 1988). Thus, I expect that the less dependent the board, the higher the probability of voluntary CEO turnover. And, one might also expect that the less dependent the board, the higher the probability of forced CEO turnover when the CEO doesn't perform well. In addition, the literature of the CEO succession issue argues that a seat on the board gives insi de directors exposure to outside directors and enables them to build social networks PAGE 30 30 and coalitions on the board (Jennings, 1971; Vancil, 1987). This development gives them more power and lends them more confidence with which to succeed the CEO. Therefore if the number of inside directors on the board is relatively small, I would expect that the firm doesn't have an optional successor for the current CEO, especially for those incumbent CEOs who are close to retirement. Thus, the probability of voluntary C EO turnover from the acquiring firm should be higher for the firm with lower board independence when there is human capital acquisition. Similarly, with lower board dependence, the effective board would procure human capital acquisition from the target fir m through merger who could perform better in the newly combined firm and thus replace the current CEO. If this is the case, under the condition of lower board dependence, the rate of the forced CEO turnover would be higher when there is top officer acquis ition. Leadership structure: The literature studies argue that the concentration of decision management and decision control in one individual reduces a firm board's effectiveness (Fama & Jensen, 1993; Jensen, 1993). To improve the efficiency of the board, it is better to separate CEO and chairman positions. Thus, when the time of retirement finally arrives, the incumbent CEO who is also the chairman on the board can be reluctant to leave his position. Or, if the incumbent CEO is far away from retirement, he/she could use his conclusive power to retain his position even when the firm's performance is poor. I expect that a powerful CEO will decrease the likelihood of CEO turnover, both voluntary and forced to step down. In addition, a CEO who is also the chai r of the board may have more power in the firm, and the firm would be more likely to depend on him/her. With new top executives entering, it is easier for them to keep the firm's dependence and power to control the firm. Accordingly, one can expect that wi th top officer acquisition, the probability of CEO turnover should be lower for PAGE 31 31 firms in which the C EO also serves as the chairman. A cquiring firms in which the CEO also serves as the chairman have less efficient board s ; hence the likelihood of CEO turnove r should be lower for firms with director acquisition from the target firm. Table 1 4 lists the mean, median, and standard deviation values of board characteristics measures for the entire sample. The mean value of board size is 11.12 for the entire sample And, on average, 27% of the directors on the board are insiders. The board size for different turnover types are close (11.38 for firms without turnover, 10.83 for firms with turnover, 11.16 for firms with voluntary turnover and 10.28 for firms with forc ed turnover.) Similarly, the mean value of inside directors is not significantly different across the subsample. The frequency with which CEOs also serve as chairman on the acquiring firm's board is high; the mean of the dummy variable is 0.69, and the median value is 1 for the full sample. The table reveal s no significant difference in leadership across subsamples. The median value of the dummy variable is one for all subsamples, both merger s with different types of turnover and without turnover. The mean value of the leadership dummy is slightly higher (0.76) for acquiring firms with voluntary CEO turnover than those acquiring firms without turnover (0.67) and acquiring firms with forced turnover (0.62). Ownership structure: The board of directors are also stockholders, thus the ownership structure should play an important role on CEO turnover. I include three variables: ownership concentration, institution ownership, and insider ownership to control the effect of ownership structure on the post merger turn over of the acquiring firm's CEO. Ownership concentration is defined as the percentage of equity held by the five largest stockholders; institution ownership is defined as the percentage of equity held by institutions; and insider ownership is the percenta ge of equity held by the officers and directors. PAGE 32 32 Ownership concentration. Economists generally suggest that there is a negative relation between the diffusion of ownership and the stockholders' incentive to monitor top managers' performance if the ownership structure is determined exogenously. Thus, I expect that the more concentrated the ownership, the more incentive the s tockholders have to monitor the CEO the greater the probability of post merger CEO turnover in the acquiring firm. Furthermore, acquiring firms with more ownership concentration have a more efficient board, thus they have more desire to acquire the valuable human capital from the target firm to succeed the current CEO who is close to retirement or does not perform well. As a resul t, I expect that with higher ownership concentration, the CEO s of acquiring firms which acquired top officers from target firm s face a higher probability of being replaced. On the other hand, with higher ownership concentration, the director acquisition should have less effect on CEO turnover since the board is more efficient. Institution ownership. Similarly the board has more incentive to effectively monitor top managers' performance if more equity is held by institutions (Smith, 1996) Thus, I expect th at the more ownership held by institutions, the greater the probability of post merger CEO turnover in the acquiring firm. Insider ownership. Morck, Shleifer and Vishny (1988) present evidence of the relationship between the shareholding of a company's ins ide directors and the firm's performance. They suggest that there are two conflicting effects of insider ownership: the positive "wealth effect" -as the number of shares held by the insiders increases, the effect on the wealth of its members from a rise in the market value of the firm increases; and the negative "entrenchment effect" -as the number of shares held by insiders increases, the likelihood of their being replaced through a proxy fight or takeover declines, and managers have more PAGE 33 33 discretion to pursue their own goals. Therefore, the direction of insider ownership's effect on the probability of CEO turnover is ambiguous. In Table 1 4, the mean value of ownership concentration, institution ownership and insider ownership, are 31.3%, 45.11%, and 11.12%, respectively. Ownership concentration is significantly higher for acquiring firms with forced CEO turnover than those without CEO turnover or with voluntary CEO turnover Institution ownership is slightly higher for acquiring firms in which the CEO i s forced to step down (47.94) than acquiring firms without CEO turnover (45.09) or with voluntary CEO turnover (43.35). Insider ownership is slightly higher for acquiring firms without CEO turnover (12.11) than acquiring firms in which CEOs are retired(9.7 1) or forced to depart (10.37), which is consistent with Denis and Sarin's (1997) results and suggests that the negative "entrenchment effect" is larger than the positive "wealth effect." Empirical Results Logit Estimates of CEO T urnover I estimate several logit regression models to test my hypothesis after controlling other explanatory variables associated with CEO turnover, in which the dependent variable is the probability that the acquiring firm's CEO is replaced within five years of t he merger announcement. Main independent variables I examined are officer acquisition and director acquisition that proxy the human capital acquisition from the target firm through merger activities. Other control variables includes CEO age, CEO age-square d, board size, board independence, leadership, ownership concentration, institution ownership, insider ownership, acquiring firms' performance before merger acquiring firms' performance after merger and target firms' performance before merger as discusse d in section 2. Model 1 includes all control variables but not human capital acquisition variables -officer acquisition and director PAGE 34 34 acquisition, where each corporate governance variable is included separately. In model 2, I add two human capital acquisiti on variables. In model 3 5 besides the independent variables included in model 2, the interaction terms of human capital acquisition and several corporate governance variables are used to capture the joint effect of corporate governance. Model 5 provides t he best specification for the logit regression model, which includes two human capital acquisition variables, all control variables discussed in section 2, and the interactions of officer acquisition and director acquisition with leadership, and ownership concentration. Table 1 6 reports the results from logit models. Each coefficient is estimated as the effect of the independent variables on post merger CEO turnover in the acquiring firm. Standard error s are shown in parentheses. The coefficient on officer acquisition is positive and significant in model s 3 and 5. This evidence supports the hypothesis that CEOs of the acquiring firm who made officer acquisition s from the target firm through a merger are more likely to be replaced than CEOs who don't acquire any top executive s from the target firm as new top officer s in the acquiring firm through merger. For example, in model 5 if the acquiring firm makes officer acquisition s the odds of the acquiring firm's CEO being replaced within five years after merger announcement (vs. not being replaced) increases by a factor of 5.9811. In contrast, the coefficient on director acquisition is negative and signific ant at the 0.05 level in model 5 indicating that CEOs of the acquiring firm who made director acquisition from the target firm through merger are less likely to be replaced than CEOs who don't acquire any top officers from the target firm as new board directors for the acquiring firm through merger. For example, in model 5 if the acquiring firm makes a direct or acquisition, the odds of the 11 To easily interpret the effect of the dependent variable on post merger CEO turnover, I calculated the odds ratio (OR). Given the for mula for the logit regression, prob (CEO turnover) = exp(x odds of independent variables are obtained by exponentiate the logit coefficients, OR=exp( ) PAGE 35 35 acquiring firm's CEO being replaced within five years after merger announcement (vs. not being replaced) decreases by a factor of 0.11. The coefficients on both CEO age and CEO age -squared are significant in all of the model s, indicating that the probability of the acquir ing firm's CEO turnover after a merger will decrease with age until a CEO reaches his/her middle age12, at which point the probability rate of being replaced increases. No significant association exists betwee n the likelihood of post merger turnover of the acquiring firm's CEO and both the acquiring firm and target firm's performance13. The coefficients on board characteristics measures, board size, board independence and leadership, are negative, but neither is sign ificant in any models. None of the intersection terms of human capital acquisition and board characteristics is significant in any model, indicating that the relation between human capital acquisition and the likelihood of CEO turnover in the acquirin g firm is unrelated to the board characteristics, i.e., acquiring firms choose their board structure either optimally or ineffectually. The coefficient on ownership concentration is positive, while the coefficient on insider ownership is negative; both are significant. The coefficient on institution ownership is negative but not significant in any model. The evidence shows that higher ownership concentration is associated with a higher probability of CEO turnover, and higher insider ownership is as sociated with a lower probability of CEO turnover in the acquiring firm while institution ownership has little effect on the probability of CEO turnover. Almost all interaction terms of 12 For example, in model 5, I calculated that the middle age for CEOs in the acquiring firm is 46 years old. 13 I replicate the analyses by substituting various measures of accounting performance, industry adjusted return on assets, return on average assets, and return on average equity. I find t hat neither before nor after merger accounting performance of acquiring firm and pre merger accounting performance of target firm is significantly related to the likelihood of CEO turnover in the acquiring firm post merger PAGE 36 36 human capital acquisition and ownership structure variables enter the models with insignificant coefficients. Only the interaction of director acquisition and ownership concentration is consistently positive and significant, indicating that with higher ownership concentration, the negative effect of director acquisition from the target firm on post merger CEO turnover in the acquiring firm would decrease. Multinomial Logit Estimates of CEO Turnover Type To examine in more detail whether the effect of human capital acquisition from the target firm is different for different ty pes of CEO turnover in the acquiring firm after merger I estimate th e multinomial logit (MNL) model14. Considering three possible types of CEO subsequent turnover, this study utilizes a three state MNL model. Using "no CEO turnover" observations as the ref erence group, the MNL model can be described as follows: State 0 : no CEO turnover State 1 : voluntary CEO turnover State 2 : forced CEO turnover 12 1 12 2 121 (no CEO turnover) 1exp() exp() (voluntary CEO turnover) 1exp() exp() (forced CEO turnover) 1exp() prob xx x prob xx x prob xx 1 is a k 1 vector of estimated coefficients for voluntary CEO turnover observations, 2 is a k 1 vector of estimated coefficients for for ced CEO turnover observations; x is a k 1 vector of explanatory variables which may influence the probability of voluntary CEO turnover or forced CEO turnover according to the prior emp irical research on CEO turnover and PAGE 37 37 merger issues, including human capital acquisition variables, the acquiring firm's CEO's characteristics, the acquiring and the target firm's characteristics, the acquiring firm's corporate governance, and year dummies. I estimate several multinomial logit models (MNL) to study different types of CEO tur nover in the acquiring firm post merger The dependent variable is type s of CEO turnover in the acquiring firm, namely, the acquiring firm's CEO has no turnover, departs voluntarily, or is forced to step down within five years of merger Independent variables are the same as in logit estimates, which include CEO age, corporate governance and firms' performance. Model 1 includes all independent variables except two human cap ital acquisition variables, where each corporate governance variable is included separately. In model 2, I add two human capital acquisition variables to model 1. In model s 3 5, besides the independent variables included in model 2, the interaction terms o f two human capital acquisition variables and several corporate governance variables are used to capture the joint effect of corporate governance. Model 5 provides the best specification for the MNL estimates, which includes all control variables and the i nteractions of officer acquisition and director acquisition with board dependence and ownership concentration. Table 1 7 contains the estimation results for model s 1 to 5 for the full sample. Each coefficient shown in Table 1 7, whether for "voluntary CEO tur nover" or "forced CEO turnover," is interpreted as relative to the omitted outcome "no CEO t urnover," and standard error is shown in parentheses. Table 1 7 has two panels: the upper half of the table reports the coefficients for the independent variable s of the likelihood of voluntary CEO turnover as relative to no CEO turnover; and the lower half reports the coefficients for the in dependent 14 The multinomial probit model g ives similar results, thus only the multinomial logit model is discussed in this paper. PAGE 38 38 variables of the likelihood of forced CEO turnover as relative to no CEO turnover. Overall chi squares and log lik elihood for five models are reported in Table 1 7; all indica te very strong model significance (p<0.001). The coefficient on officer acquisition for voluntary CEO turnover is positive, but only significant in model 5 (p<.05) in which I included several in teraction terms. To interpret the effect of independent variables better, similarly as in the logit model, I calculate relative risk ratio ( RRR) of independent variables on the post merger CEO turnover15. For example, in model 5, for acquiring firms with officer acquisition relative to those without officer acquisition, the relative risk for a CEO being replaced voluntarily within five years after merger relative to a CEO who doesn't leave would be expect ed to increase by a factor of 11.78, given that the other variables in the model are held constant. In other words, acquiring firms who acquired top executives from the target firm as new top managers in the acquiring firm after merger are more likely to r eplace their CEO voluntarily over no turnover than those who didn't acquire any human capital. The coefficient on the officer acquisition for forced CEO turnover is positive and significant in almost all of the models. The only model in which the coefficie nt of officer acquisition is not significant is the one where no interaction terms have been included. The relative risk for a CEO being fired as opposed to a CEO who doesn't leave is expected to increase by a factor of 15.44 for acquiring firms with offic er acquisition relative to those without officer a cquisition, which is larger than the effect of top officer acquisition on voluntary CEO turnover. That is, acquiring firms who acquired top executives from the target firm as new top managers in the acquiri ng firm after merger are more likely to fire their CEO over no 15 The RRR of a coefficient indicates how the risk of the outcome falling in the comparison group compared to the risk of the outcome falling in the refe rence group (in my case, no CEO turnover group) changes with the variable in question, which is similar to the odds ratio in the logit model. PAGE 39 39 turnover than those who didn't acquire any human capital. The evidence provides support for hypothesis 1 which states that if the target firm's top executive s are retained as top executives in the merged entity after the merger the acquiring firm's CEO is more likely to leave, by both voluntary and forced out. The coefficient on director acquisition is negative for both voluntary and force d CEO turnover. The negative impact of director acquisit ion for voluntary CEO turnover is significant in model s 4 and 5 in which I included interaction terms of human capital acquisition and ownership concentration and board independence, indicating that acquiring firms who acquired top executives from the target firm as new board directors in the acquiring firm after merger are less likely to replace their CEO voluntarily over no turnover than those who didn't acquire any ne w top officer. The results show that for acquiring firms with director acquisition relat ive to those without director acquisition, the relative risk for a CEO being replaced voluntarily within five years after merger relative to a CEO who doesn't leave would be expected to increase by a factor of 0.22 in model 4 and 0.16 in model 5, given the other variables in the model are held constant. For forced CEO turnover, the coefficient on director acquisition is negative but not significant in most of models. It is only significant in model 4, in which I included interaction terms of human capital a cquisition and ownership concentration. The results indicate that acquiring firms who acquired top executives from the target firm as new board directors in the acquiring firm after merger are less likely to fire their CEO over no replacement than those wh o didn't acquire any new director. Thus, for acquiring firms with director acquisition relative to those without director acquisition, the relative risk for a CEO being forced out within five years after merger relative to CEO who doesn't leave would be expected to increase by a factor of 0.1 as shown in model 4, given the other variables in the model are held constant. The insignificant PAGE 40 40 effect of director acquisition could be explained by the optimal board structure in the acquiring firm. Hence, CEO s have little effect on the board structure; new directors from the target firm will monitor the incumbent CEO, and provide advice to strategy making team s to perf orm efficiently. Therefore, director acquisition through merger is an efficient decision for the acq uiring firm, and it has little effect to force CEO to step down. Table 1 7 also reports influences of other rele vant variables. The coefficient for CEO age is consistently negative, and is consistent ly positive for CEO age squared; both are significant in all 5 models for both voluntary turnover and forced CEO turnover. The results indicate that the effect of CEO age on CEO turnover is curvilinear. In model 5, I find that the likelihood of CEO turnover decreased until the age of 45 for CEOs with voluntary t urnover and 48 for CEOs with forced turnover, when it begins to increase. None of the firm's performance variables is significant for voluntary CEO turnover. For forced CEO turnover in the acquiring firm, both acquiring firm and target firm's performance a round merger are insignificant; only acquiring firm's ROE before merger is slightly positive and only significant in model 5. None of the corporate governance variables are significant for voluntary turnover of the acquiring firm's CEO, indicating that vol untary CEO turnover in the acquiring firm is unrelated to corporate governance. The coefficient on ownership concentration is positive and significant for forced CEO turnover in almost all of the models; all other corporate governance variables are not ver y significant for forced CEO turnover. When the interaction between corporate governance variables and human capital acquisitions are included in the model, for voluntary CEO turnover, the coefficient for the interaction of director acquisition and ownersh ip concentration is positive and significant (p<0.01), and the coefficient for the interaction of director acquisition and board independence PAGE 41 41 is negative and significant (p<0.1). For forced CEO turnover, only the coefficient on the interaction of director acquisition and ownership concentration is positive and significant (p<0.01) ; none of the other interaction terms enter the models with significant coefficients. Conclusion Th is study adds a dimension of understanding heretofore overlooked in the literature regarding the effect of a merger on the leadership change in the acqui ring firm after the merger, and suggests that one of the motivations for merger is to acquire talented human ca pital. Based on a sample of 236 completed mergers from 1996 through 2000, the evidence from this study is quite supportive of the view that human capital acquisition from the target firm through merger influence s post merger turnover of the acquiring firm' s CEO. Talented human capital from the target firm will change both top management team and board structure of the acquiring firm, and thus influence leadership change in the acquiring firm. If the target firm's top executive is retained as top executive in the newly combined firm after merger the acquiring firm's CEO is more likely to leave, either voluntarily or involuntarily In contrast, if top executives of the target firm are retained as new board directors in merged entity post merger the acquirin g firm's CEO is less likely to leave voluntarily, but no change occurs in the probability of being forced out I find no significant association between the acquiring firm's board characteristics and the probability of CEO turnover Finally, only ownership concentration has a slightly positive effect on the relation of director acquisition and CEO turnover in acquiring firm after merger; other ownership structure variables are not related to the probability of post merger turnover of the acquiring firm's CE O. In addition, my study complements Shleifer and Vishnys (2003) theory of stock market driven acquisition. There is an additional reason why there are so many merger s Shleifer and Vishny's theory states that the irrational overvalued acquiring firm's equity is the reason behind PAGE 42 42 many acquisitions. For instance, the acquiring firm acquires firms with less -valued assets by using overvalued stock. However, this study suggests that acquiring valuable human capital is an important reason to implement merger although such human capital acquisition through merger may affect the probability that CEOs who make mergers are replaced. This study represents the first attempt to empirically examine the relation between human capital acquisition from the target firm through merger and post merger leadership change in the acquiring firm. Though I believe that the evidence gained is valuable, there are several limitations. First, I am unable to directly observe and examine the direct human interactions surrounding the p rocess of human capital acquisition on post merger CEO turnover in the acquiring firm. Second, the study is appropriate mainly for mergers or acquisitions when the deal is la rge enough, and therefore the acquiring firm is more likely to gain human capital acquisition. This argument may be not good for small merger and acquisition activities. Several extensions of this study could make significant contributions to the literature. First, a study of CEO successorship in the acquiring firm would be helpful. It would be useful to understand whether top executives acquired from the target firm take CEO position after a merger, and if not, who is named as heir apparent and what happens should they fail to succeed. Second, we could better understand the process of l eadership change associated with human capital acquisition from the target firm by examining turnover of lower level top management i.e. president, chairman of the board, vice presidents, CFO and COO. These issues are beyond the scope of this study, but th ey may provide advance understanding of the effect of a merger on the leadership changes within the acqui ring firm PAGE 43 43 Table 1 1 Sample d istributio n and frequency of the CEO turnover Panel A. Distribution by Turnover Type Type No. of M&As % of sample % of subsample Total sample 2 36 100 CEO Turnover 1 09 46 100 Voluntary CEO Turnover Forced CEO Turnover 67 28 61 42 18 39 No CEO Turnover 1 27 54 Panel B. Distribution by Year Year No. of M&As No. of t urnover (%) No. of no turnover (%) No. of voluntary turnover (%) No. of forced t urnover (%) 1996 37 20 (54 %) 17 (46 %) 15 ( 41 %) 5 ( 13 %) 1997 47 17 (36 %) 30 (64 %) 11 (23.53%) 6 (13 %) 1998 55 21 (38 %) 34 (62 %) 13 (23.44%) 8 (15 %) 1999 59 33 (56 %) 26 (44 %) 16 (25.76%) 17 (29 %) 2000 38 18 (47 %) 20 (53 %) 12 (30.95%) 6 (16 %) Note: The sample consists of 236 completed M&As between two public companies during the 19962000 period. This table reports the frequency of the CEO turnover across different years and types. PAGE 44 44 Table 1 2. Descriptive statistics for human capital a cquisition Officer A cquisition Director A cquisition Total Sample Mean 0.2839 0.3475 Std. Dev. 0.4518 0.4772 No CEO T urnover Mean 0.2441 0.3071 Std. Dev. 0.4313 0.4631 CEO T urnover Mean 0.3303 0.3945 Std. Dev. 0.4725 0.491 0 V oluntary CEO Turnover Mean 0.2836 0.3881 Std. Dev. 0.4541 0.491 0 Forced CEO Turnover Mean 0.4048 0.4048 Std. Dev. 0.4968 0.4968 PAGE 45 45 Table 1 3 Variable de finitions Data from SDC M&A database, COMPUSTAT and SEC database from 1996 to 2000 Dependent Variable CEO turnover: 0 if no CEO turnover in the acquiring firm 1 if voluntary CEO turnover in the acquiring firm, 2 if forced CEO turnover in the acquiring firm Independent Variables Human capital acquisition: Of ficer acquisition: 1 if there is one or more top managers of the target firm have retained position s in the merged entity and reported as top management executive after merger effected, and 0 otherwise. Director acquisition: 1 if there is one or more top managers of the target firm beca me a director in the merged entity board after merger effected, and 0 otherwise. Other Control variables: CEO age: age in years CEO age -squared: age2 Board size: the number of directors reported in board. Board independence: the percentage of inside directors on the acquiring firm 's board at the year of the merger announcement. CE O/Chairman: 1 if the CEO of the acquiring firm also serves as chairman of the board, 0 otherwise Ownership concentration: the percentage of equity held by the five largest blockholders. Insider ownership: the percentage of equity held by officers and directors. Institutions ownership: the percentage of equity held by institutions. Pre -ROE of acquirer : the industry adjusted yearly return on the common equity of the acquiring firm in the year before the merger announcement. Post -ROE of acqui rer : the industry adjusted yearly return on the common equity of the acquiring firm in the year right after the merger announcement. Pre -ROE of target: the industry adjusted yearly return on the common equity of the target firm in the year before the merger announcement. Interaction terms (with o fficer acquisition or d irector a cquisition) Officer acquisition Board independence Officer acquisition CEO/Chairman Officer acquisition Ownership concentration Director acquisition B oard independence Director acquisition CEO/Chairman Director acquisition Ownership concentration PAGE 46 46 Table 1 4 Explanatory variable descriptive statistics for the total sample Variable Mean Std. Dev. Median Min Max CEO age 53. 85 59 7.6 2 89 54 37 83 Board size 11. 12 29 4. 545 5 10 2 28 Board independence 0.2 70 1 0.17 5 1 0.2 5 0 1 CEO/Chairman 0. 686 4 0.4 6 49 1 0 1 Ownership concentration 31.299 5 2 8.14 85 2 4.34 0 99.99 Institution ownership 45. 105 1 27.2 0 18 48.92 0 99. 86 Insider ownership 1 1.1 199 16. 438 4 4.125 0 99.99 Pre ROE of acquirer 5.57 2 4 63. 73 49 3. 16 146.435 908.5188 Post ROE of acquirer 7.70 55 133.881 8 1.874 427.347 1858.282 Pre ROE of target 28 75 66 402.47 04 0 6170.87 195.7014 PAGE 47 47 Table 1 5 Correlation s Turnover Turnover type Officer a cquisition Director acquisition CEO age Board size Board independence CEO/ Chairman Turnover 1 Turnover type 0.9019* 1 Officer a cquisition 0.0953 0.1243* 1 Director acquisition 0.0915 0.0876 0.5274* 1 CEO age 0.3197* 0.1932* 0.0189 0.0734 1 Board size 0.0607 0.0824 0.0306 0.1568* 0.0408 1 Board independence 0.0321 0.0192 0.1156* 0.0057 0.0746 0.3108* 1 CEO/Chairman 0.0399 0.0078 0.1015 0.0520 0.1799* 0.0481 0.0428 1 Ownership concentration 0.1384* 0.1987* 0.797 0.1046 0.1165* 0.2182* 0.0798 0.0083 Institution ownership 0.0005 0.0246 0.1067 0.0310 0.0125 0.0580 0.0605 0.2476* Insider ownership 0.0653 0.0531 0.0884 0.0623 0.0858 0.1740* 0.0794 0.0698 Pre ROE of acquirer 0.0550 0.0807 0.1174* 0.0727 0.0275 0.0084 0.1550* 0.1275* Post ROE of acquirer 0.0864 0.1342* 0.0098 0.0456 0.0739 0.0241 0.0183 0.0442 Pre ROE of target 0.0607 0.0520 0.0444 0.0449 0.0351 0.1246* 0.0612 0.0467 Ownership concentration Institution ownership Pre ROE of acquirer Post ROE of acquirer Pre ROE of target Ownership concentration 1 Institution ownership 0.1879* 1 Insider ownership 0.3476* 0.1075* Pre ROE of acquirer 0.0142 0.0844 1 Post ROE of acquirer 0.0226 0.0155 0.0479 1 Pre ROE of target 0.0614 0.1177* 0.0151 0.0158 1 PAGE 48 48 Table 1 6 Re sults of Logit Regression Model 1 2 3 4 5 Officer acquisition 0.4745 1.4689* 1.0472 1.7878* (0.3935) (0.8488) (0.6791) (1.0279) Director acquisition 0.0589 0.7411 1.6849** 2.226** (0.3758) (0.7586) (0.6609) (0.9726) CEO age 0.7902** 0.8051** 0.7723** 0.7618** 0.732** (0.3121) (0.3144) (0.3189) (0.3258) (0.3307) CEO age 2 0.0084 *** 0.0085 *** 0.0083 *** 0.0082 *** 0.0080* (0.0030) (0.0030) (0.0030) (0.0031) (0.0031) Board size 0.0105 0.0122 0.0226 0.0099 0.0176 (0.0364) (0.0373) (0.0385) (0.0392) (0.0403) Board independence 0.3124 0.4347 0.4971 0.6756 0.6877 (0.9325) (0.9405) (0.9523) (0.9621) (0.9714) CEO/Chairman 0.0631 0.1266 0.1144 0.0542 0.0477 (0.3345) (0.3395) (0.4185) (0.3495) (0.4207) Ownership concentration 0.0204 *** 0.0195 *** 0.0198 *** 0.0094 0.0091 (0.0063) (0.0064) (0.0064) (0.0081) (0.0081) Institution ownership 0.0046 0.0052 0.0051 0.0063 0.006 (0.0062) (0.0062) (0.0063) (0.0065) (0.0066) Insider ownership 0.0215* 0.0207 0.0219 ** 0.023** 0.0228 (0.0108) (0.0108) (0.0110) (0.0117) (0.0117) Pre ROE of acquirer 0.0029 0.0024 0.0023 0.0023 0.0021 (0.0026) (0.0026) (0.0028) (0.0026) (0.0027) Post ROE of acquirer 0.0018 0.0018 0.0019 0.0018 0.002 (0.0021) (0.0020) (0.0021) (0.0020) (0.0021) Pre ROE of target 0.001 0.0008 0.0006 0.0008 0.0007 (0.0037) (0.0027) (0.0019) (0.0023) (0.0018) Officer acquisition CEO/Chairman 1.2765 1.0625 (0.9624) (1.0623) Director acquisition CEO/Chairman 0.8778 0.8041 (0.8683) (0.9824) Officer acquisition ownership concentration 0.0194 0.0162 (0.0171) (0.0169) Director acquisition ownership concentration 0.0508 *** 0.048*** (0.0169) (0.0166) Constant 17.896 ** 18.303 ** 17.479 ** 17.242 ** 16.46* (8.1066) (8.1704) (8.2943) (8.4658) (8.6012) Model Chi2 57.07*** 58.80*** 60.69*** 70.77*** 71.8 6*** Log Likelihood 134.361 133.497 132.55 127.51 126.968 *, ** and *** indicate significance at 10%, 5% and 1% levels, respectively. PAGE 49 49 Table 1 7. Results o f multinomial logistic regression m odel 1 2 3 4 5 Equation 1: Voluntary CEO Turnover Officer acquisition 0.1853 0.2589 0.7728 2.4661** (0.4656) (1.1125) (0.7709) (1.1953) Director acquisition 0.0442 0.3388 1.5131** 1.8500* (0.4356) (0.8818) (0.7275) (1.0973) CEO age 0.7894** 0.8079** 0.8262** 0.7647** 0.7646** (0.3507) (0.3507) (0.3569) (0.3625) (0.3712) CEO age squared 0.0086*** 0.0088*** 0.009*** 0.0085** 0.0085** (0.0033) (0.0033) (0.0034) (0.0034) (0.0035) Board size 0.0086 0.0074 0.0063 0.0119 0.0105 (0.0416) (0.0426) (0.0442) (0.0442) (0.0449) Board independence 0.0818 0.0279 0.0427 0.1772 1.3122 (1.0561) (1.0696) (1.0820) (1.0879) (1.4145) CEO/Chairman 0.1478 0.1173 0.034 0.174 0.2898 (0.3972) (0.4014) (0.4861) (0.4095) (0.4174) Ownership concentration 0.0146* 0.0138* 0.0146* 0.0034 0.0004 (0.0075) (0.0076) (0.0076) (0.0097) (0.0098) Institution ownership 0.0083 0.0085 0.0084 0.0101 0.0103 (0.0071) (0.0072) (0.0072) (0.0074) (0.0077) Insider ownership 0.0178 0.0172 0.017 0.0194 0.0175 (0.0123) (0.0124) (0.0124) (0.0132) (0.0130) Pre ROE of acquirer 0.0014 0.0012 0.0015 0.001 0.0024 (0.0038) (0.0038) (0.0041) (0.0038) (0.0039) Post ROE of acquirer 0.0004 0.0005 0.0005 0.0009 0.001 (0.0032) (0.0033) (0.0033) (0.0036) (0.0037) Pre ROE of target 0.0102 0.0106 0.0135 0.0114 0.012 (0.0072) (0.0074) (0.0083) (0.0076) (0.0077) Officer acquisition CEO/Chairman 0.1134 (1.2291) Director acquisition CEO/Chairman 0.5326 (1.0026) Officer acquisition ownership concentration 0.0215 0.0236 (0.0199) (0.0203) Director acquisition ownership concentration 0.0528*** 0.0590*** (0.0192) (0.0198) PAGE 50 50 Table 1 7 Continued Officer acquisition board independence 1.3008 (3.4754) Director acquisition board independence 6.3018* (3.6288) Constant 16.6395* 17.1595* 17.6774* 16.1104* 15.8288 (9.2408) (9.2400) (9.3901) (9.5460) (9.7505) Equation 2: Forced CEO Turnover Officer acquisition 0.7799 2.5457** 1.6189* 2.7367** (0.5088) (1.0997) (0.8953) (1.2689) Director acquisition 0.2155 1.379 2.2698** 1.1546 (0.4977) (1.0632) (0.9460) (1.2561) CEO age 0.6853* 0.7244** 0.6596* 0.6774* 0.6770* (0.3629) (0.3659) (0.3760) (0.3797) (0.3874) CEO age squared 0.0071** 0.0075** 0.0069* 0.0071** 0.0071* (0.0034) (0.0035) (0.0036) (0.0036) (0.0037) Board size 0.0454 0.047 0.0672 0.0497 0.0748 (0.0531) (0.0535) (0.0563) (0.0572) (0.0607) Board independence 1.0406 1.1603 1.2324 1.6118 0.6635 (1.3438) (1.3544) (1.3956) (1.4019) (1.6575) CEO/Chairman 0.329 0.4341 0.2348 0.3645 0.3392 (0.4384) (0.4470) (0.5789) (0.4588) (0.4691) Ownership concentration 0.0270*** 0.0263*** 0.0279*** 0.0172* 0.0143 (0.0076) (0.0078) (0.0080) (0.0102) (0.0103) Institution ownership 0.0002 0.0012 0.0012 0.0017 0.0015 (0.0079) (0.0080) (0.0082) (0.0084) (0.0086) Insider ownership 0.0223 0.021 0.0249* 0.0244* 0.0233 (0.0139) (0.0139) (0.0145) (0.0147) (0.0146) Pre ROE of acquirer 0.0037 0.0032 0.0029 0.0032 0.0069** (0.0027) (0.0027) (0.0028) (0.0027) (0.0032) Post ROE of acquirer 0.0034 0.0031 0.0038 0.003 0.0028 (0.0031) (0.0031) (0.0033) (0.0031) (0.0031) Pre ROE of target 0.0002 0.0001 0.0001 0.0002 0.0001 (0.0011) (0.0010) (0.0010) (0.0010) (0.0010) Officer acquisition CEO/Chairman 2.3608* (1.2604) PAGE 51 51 Table 1 7. Continued Director acquisition CEO/Chairman 1.4673 (1.2005) Officer acquisition ownership concentration 0.0255 0.0267 (0.0200) (0.0206) Director acquisition ownership concentration 0.0575*** 0.0710*** (0.0207) (0.0223) Officer acquisition board independence 6.4915 (4.0871) Director acquisition board independence 3.6506 (3.1609) Constant 15.314 16.3031* 14.6181 15.2263 15.1151 (9.5271) (9.6190) (9.8955) (9.9759) (10.1600) Pseudo R Square 3.173*** 3.179*** 3.188*** 3.207*** 3.227*** Model Chi2 81.768 84.348 88.792 97.347 106.853 Log likelihood 194.672 193.382 190.532 186.882 182.129 *, ** and *** indicate significance at 10%, 5% and 1% levels, respectively. PAGE 52 52 CHAPTER 2 ESTIMATIO N OF CENSORED REGRESSION MODEL: A SIMULATION STUDY Introduction Censoring is common in econometric applications. Censoring usually occurs for two reasons. First, censoring is the result of individual rational behavior subject to a non negativity constraint. For instance, in empirical analysis of individual labor supply, hours worked are positive if and only if the individual chooses to participate in the labor force (e.g. Heckman, 1979, 1980). Lik ewise, the observed consumption of meat for example, is positive if and only if the individual chooses to consume meat. F irm 's R&D expenditure is positive if and only if the firm engages in R&D activity. Second, censoring is the result of survey design. For example, in data collection, it is common practice to top and/or bottom code the income variable. In this case, the empiri cal analysis of income requires a model to deal with censoring data problem (see Solon 1992, 1999; Zimmerman 1992; Ashenfelter and Zimmerman 1997, for application). A p opular model to deal with censoring data problem is Tobit model. The standard approac h for e stimating the Tobit model is maximum likelihood estimation. For panel data with censoring data problem, a natural choice of models is the panel data Tobit model with individual effect. However, estimation of the panel data Tobit model, when the indi vidual effect is allowed to cor relate with the explanatory variables arbitrarily, is nontrivial and difficult. Under some conditions, Honor (1992) derives a conditional moment restriction which requires discarding part of observations. Based on this condi tional moment restriction, Honor proposes a consistent and asymptotically normally distributed estimator. His estimator, however, is not efficient since it does not use all moment restrictions implied by the conditional moment restriction. To increase the efficiency of the estimates, one could exploit other moment restrictions by applying the two-step GMM, or the continuously updating GMM, or the empirical likelihood estimator (ELE). Asymptotically, the latter three PAGE 53 53 estimators are equivalent and are more e fficient, at least not worse, than the Honor estimator. The main objective of this paper is to study two issues. First, I consider the efficiency of these estimators in large samples. It is interesting to see how much efficiency can be gained by the last three estimators relative to the Honor estimator when sample size is moderate or large. Second, we argue the small sample properties of all four estimators, particularly the relative performance of the Honor estimator vs. the other three estimators. The remainder of the paper is organized as follows. Section 2 sets up the model and describes the four estimators. The virtues and drawbacks of the estimators are discussed. Section 3 studies the finite sample performances of these estimators via a Monte Carlo study and discusses the Monte Carlo results. Section 4 concludes. Model and Estimators The standard model for cen sored dependent variable is Tobit model proposed by Tobin (1956). The standard approach for extracting the rich information in panel data is to use the individual specific effect (see Hsiao, 1986). Thus, a natural model for analyzing a panel dataset containing censored dependent variables is the panel data Tobit regression model with individual effects given by 0,max0,,1,2,....,;1,2,....,itiitit it ityx yyiNtT (2 1) where i denotes the individual, t denotes time; ity denotes the lat ent dependent variable; ity denotes the observed dependent variable; itx denotes the k -dimensional column -vector of explanatory variables; i denotes the unobserved individual specific effect; 0 denotes the true value of the unknown parameter (column) vector to be estimated; and it denotes the error term. The error term it is usually assumed to be normally distributed; and in this case the model is called typ e I Tobit model (see Amemiya 1985, for other types of Tobit models). PAGE 54 54 Estimation of Tobit model particularly the version of model (2 1 ), can be difficult, depending on the assumptions imposed on the error term it and the individual effect i If we as sume the individual effect in model (2 1 ) is random, where both the error term and the individual effect are assumed to be uncorrelated with the explanatory variables and normally distributed, then the standard maximum likelihood estimator (MLE) of model (2 1 ) is consistent and asymptotically normal. The normally distributed error term, though commonly imposed in empirical ana lysis, is difficult to justify. Moreover, the assumption that the individual effect is independent of the explanatory variables is equally difficult to justify Without independence between the individual effect and the explanatory variables, estimation of model (2 1 ), requires using the fixed effect Tobit model, which is difficult even if the normality assumption on the error term is maintained. Without both the independence between the individual effect and the explanatory variables and the parametric spe cification of the error term density, estimation of model (2 1 ) is even more difficult. The difficulty arises because the individual effect i enters the model nonlinearly and the simple time -differencing approach widely used in lin ear panel data models does not work here. To see thi s, notice that, for any period t at the true value 0 the residual value is 0 00 0max,max,itit ititit iitityxyxx x (2 2) which clearly is censored at 0 itx Thus, for any two periods t and s at true value 0 applying simple time differencing, we obtain: 00 0 0max,max,itit isis iitit iisisyxyx x x (2 3) The unobserved individual effect i is obviously not removed here. But this does not necessarily mean that standard regression techniques do not yield consistent estimates for the model parameter. Standard regression techniques still produce a consistent estimate i f the PAGE 55 55 right hand side of ( 2 3) is uncorrelated with the explanatory variables or some instrumental variables1. Unfortunately, the right hand side of ( 2 3) is correlated with the explanatory variables and it is hard to find instrumental variables. To overcome this problem, Honor (1992) suggests a clever transformation of the model so that the explanatory variables are uncorrelated with the time -differenced residuals. The intuition of his approach is that the linearity of the model holds for those individuals whose dependent variables are not censored in both periods and are not far apart. Applying simple time differencing to those individuals would preserve the zero correlation between the explanatory variables and the residu als. Specifically, define the artificially censored residuals as 0 00 00()max,max,,titit ititis iititiseyxyxx xx 0 00 00()max,max,,sisis isisit iisisiteyxyxx xx The following assumption is imposed: Assumption : The error terms it and is conditional on (,,)itisixx are iden tically distributed. Under the above assumption, it is easy to show that (2 4) holds for any function () q To see exactly which observations are used to determine the parameters, denotes iitisxxx For the sake of arguments, suppose that 00ix Consider four cas es: (i) the dependent variable is not censored in both periods; (ii) the dependent variable is censored in both periods; (iii) the dependent variable is censored in period t but not in period s ; and (iv) the dependent variable is not cens ored in period t but censored in period s 1 Therefore, the conditional mean given the explanatory variables is zero. 00(())(()),0titit sisisitisEqeyxqeyxxx PAGE 56 56 Case (i) : 0,0itisyy In this case, we have 00()isis isiseyxyx 00 0 00, if () if is itit itit itit ititxyx eyx yxyx So trimming occurs if 0 itiyx Regardless of w hether trimming is used or not, observations in this case are used to determine the model parameters. Case (ii) : 0,0itisyy In this case, it is straightforward to show that 00()()itit isiseyxeyx Hence, observations like these are not used in conditi onal moment restrictions ( 2 4) to determine the model parameters. Case (iii) : 0,0itisyy In this case, we have 00()itit iseyxx 00()isis isiseyxyx Observations like these are used to determine the model parameters. Case (iv): 0,0itisyy In this case, we have 00()isis iteyxx 00 0 00, if () if is itit itit itit ititxyx eyx yxyx Clearly, trimming occurs when 0 itiyx When tri mming occurs, the observation cannot be used to determine the model parameters. PAGE 57 57 To summarize, the co nditional moment functions (2 4 ) only uses observations where the dependent variable is not censored in both periods as well as observations where the dependent variable is censored in one period but not in the other period and where no trimming occurs. Figure 2 1 and Figure 2 2 show the set of observations that has been discarded in estimation. Now, we present several estimators discussed in the literature which use conditional moment restriction s (2 4) to obtain consistent estimations for panel data Tobit model with fixed effect. Honor Estimation Based on the conditional moment restrictions (2 4 ), Honor (1992) proposes four estimators. In this study, we focus on the last estimator, which has a conv ex objective function2 and the first order condition that coincides with some unconditional moment restrictions (wh ich given in (2.5) in Honor, 1992) By using his setting for the case of two time periods (i.e., 1,2 st ), we have the following estimator: 1 argmin(, )N H itisi istyyx (2 5 ) where N is the number of observations, and ix 2 1 1112 2 2 2 1 2121 2 2 1 22122 2, if (, )(), if < if < itisyyyyy yyyyyy yyyyy Under sufficient conditions, Honor shows that H is consistent and asymptotically normally distributed: d 11 0 ()(0,)HNNV w here 2Thus it is straightforward to derive the limit distribution of the estimator. The corresponding conditional moment restriction will be referred as "smooth" conditional moment restriction. PAGE 58 58 1 21 1 1,N H iii N iyxyxx 22 21 1 2 2 1 1202 1 11 1 ( )1HH N iiiiii H i iiiiiiyyxyxy V xx N yyxyxy GMM E stimation Noting that the Honor estimator does not use all information implied by conditional moment restrictions (2 4 ), Ai and Li (2006) suggest that other moment restrictions implied by conditional moment restrictions (2 4 ) can be used to increase effic iency. Specifically, their idea is to use a series of basis functions3 to approximate the arbitrary function () q F or some integer 1k let 112()((),(),...,())kqzqzqzqz denote known basis functions that approximate any square integrabl e function of z where 0(),ititzeyx 1,2,..., tT For some integer 2k let 212()((,),(,),...,(,))i itisitiskitispxpxxpxxpxx denote known basis functions that approximate any square integrabl e function of ix where (,).iitisxxx Condition (2 4 ) implies the following unconditional moment restrictions: 00(())(())()0itit isis iEqeyxqeyxpx (2 6 ) where denotes the Kronecker product. Obviously, equation (2 6 ) includes 12kk moment functions. Let (,,)iigyx denote the 12kk -dimensional column vecto r of moment functions in 3Alternatively, we could approximate the conditional mean functions (2 4) by using the fitted value from the regression () qz on ()ipx More details are presented in Monte Carlo study. PAGE 59 59 (2 6 ): 00(,,)(())(())()ii itit isis igyxvecqeyxqeyxpx where 1,2,...,1 sT ,and 1,..., tsT Denote 11 11 ()(,,), ()(,,)(,,)NN ii iiii NN iiggyx gyxgyx Also, let be th e initial estimator obtained by 1argmin()() gWg where W is a random weighting matrix, and it is positive definite. Hansen's (1982) best 2 -step GMM estimator is 1argmin()()()GMMgg (2 7 ) Altonji and Segal (1996), however, document that, although Hansen's 2 -step GMM estimator has desirable lar ge sample properties, it has poor finite sample performance. Hansen, Heaton and Yaron (1996) propose a continuously updated best GMM estimator which h as smaller bias than 2 -step GMM 1argmin()()()updategg (2 8 ) The continuously updating GM M is analogous to the 2 -step GMM except that the criterion function in ( 2 8 ) is simultaneously minimized over in both 1 () and () g Newey and Smith (2004) show that the continuously updating GMM has better finite sample properties. Empirical Likelihood Estimation Alternatively, the empirical likelihood approach could be applied to conditional moment restrictions (2 4 ) (Qin and Lawless, 1994; Kitamura, Tripathi and Ahn, 2004; Donald, Imbens, and Newey, 2004). The idea here is to treat the joint density (,) fyx as unknown and estimate it together with the model parameter 0 Specifically, let i denote the PAGE 60 60 probability of (,)iiyx Then the model parameter can be estimated by the following constrained maximum likelihood estimation: ,1 1 i 11 argmaxln subject to 1, (,,)0, 0.N EL i i NN i iii iigyx (2 9 ) Since t he constrained optimization (2 9 ) is hard to implement, the estimator can be difficult to compute. Notice that by applying the Lagrange approach, it is straightforward to show that the maximum likelihood estimator solves: () 1 argminmaxln(1(,,))N EL ii igyx (2 10) where is a vector of Lagrange multipliers. EL is the empirical likelihood estimator. Newey and Smith (2004) also show that for quadratic ()ipx update is identical to ,1EL in their theorem 2.1. The five estimators presented above, though all derived from the same conditional moment restriction s (2 4), have different computational advantages and disadvantages. Honor's estimator is the easiest to com pute since his criterion function is globally convex. Hansen's best GMM estimator has desirable large sample properties, but its finite sample performance is poor. The continuously updating GMM estimator is more difficult to compute since the criterion fun ction is not globally convex. Moreover, the weighting matrix can be singular for certain values of parameters, especially when sample size is small relative to the number of moment functions in the conditional moment restrictions. When this happens, the iteration process that searches for the estimator would stop. The empirical likelihood est imator ,1EL is also difficult to compute because the constraints may not be satisfied by all parameters. The advantage is th at the implied probabi lity i is also estimated. The EL version of the empirical likelihood estimator is relatively easier to compute since it does not PAGE 61 61 force the constraints to be satisfied. The advantage of the last three estimat ors is that they are asymptotically equivalent and superior to the Honor estimator up to the sec ond order (See Newey and Smith, 2004). The disadvantage is that all three estimators are hard to compute. Monte Carlo Experiments To analyze the interesting issues mentioned above, a Monte Carlo study is conducted. In all the four designs, I consider two time periods (i.e. 2 T ) and include two explanatory variables. For each individual and each period, denote 12(,)itititxxx where 1,2 t with 1 itiitx 4. The true value of the parameter is 0(1,1) All results presented in this paper are from 1000 replications of model (2 1). Design 1 In design 1, the random variables 12,,,iii and 2 itx ( 1,2 t ) are independent of each other and are drawn from the standardized chi -squared distribution5 of degree of freedom three (with mean zero and variance one)6. Conditional on i the e rror terms 1 i and 2 i are independent and follow the normal distribution 2 11 22(0,)iN 7. Implementing GMM and EL estimators requires the specified form of the moment functions. In the simulation study, I choose the polynomial basis functions which are given by 123 1()(,,,...)kqzzzz and 4 Allow for correlation between covariates and individual effect would give more general results. 5 Normally distributed explanatory variables may give too optimistic conclusions from Monte Carlo's (Chesher, 1995). 6 Data is generated by Matlab 7.0.4. By using "seed=11", we assure that all 1000 replications are the same for different sample size N and 12(,) kk so that it is meaningful to compare the results. 7 Without loss of generality, we allow the heteroskedasticity by letting the variance of the error terms depends on the fixed effect. PAGE 62 62 212112221122212111()(,,,,1,,...)iiiiiiiiikpxxxxxxxxx where () qz is a 1k -d imensional column -vector of polynomial basis functions of z, a nd ()ipx is a 2k dimensional column-vector of pol ynomial basis functions of ix For instance, when 11 k and 22 k the unconditional moment restrictions ( 2 5) can be written as 2220111021[(,,)]{[()()]()}0ii iiiiiiEgyxEeyxeyxxx (2 11) In the appendix, I show that when 11 k and 22 k the moment functions ( 2 6) are the same moment functions used in the Honor model8. For other 1k and 2k the m oment functions are constructed analogously. By changing 12(,) kk different numbers of conditional moment functions can be used to estimate the model. To evaluate the small sample properties, we consider N=200. For each sample we compute estimators when 12(,) kk takes on the following values: (1, 2), (1, 5), (1, 9), (1, 1 5), (2, 5), (2, 9), (2, 15), (3, 5), (3, 9), (3, 15)9. To evaluate efficiency in the large sample, we consider N=500 and N=100010. Design 2 Design 2 differs from design 1 only in the distribution of the regressors. The random variables 12,,,iii and 2 itx ( 1,2 t ) are independent from each other and are drawn from the standardized chi -squared distribution of degree of freedom one (with mean zero and 8 In order to compa re with the Honor estimator, (,,)iigyx used in the continuously updating GMM should include the moment functions used in the Honor model. Based on Honors design, I show in appendix A that 11 k and 22 k gives exactly the same moment functions as those of the Honor model. 9 Considering the sensitivity to the choice of polynomial, we use only second order self power series for 29 k and all second order power series for 215k 10 For N=200, 500, 1000, when 12(,) kk is large, there are some observations cannot obtain converged GMM estimator, thus we can't include them in our sample. Therefore, we discard the H Honors results obtained from these bad observations, and generate additional number of samples for both the Honor estimator and the updating GMM to ensure the comparability. PAGE 63 63 variance one). By decreasing the degree of freedom of the chi -squared distribution, design 2 generates samples with small variance. Design 3 In design 3, I modify design 1 by increasing the noise in the error terms. Conditional on ,i the error terms 1 i and 2 i are independent and follow the normal distribution 2(0,1)iN Design 4 Instead of using the transformed unconditional moment functions ( 2 5), in design 4, I use the conditional moment functions ( 2 4), where the condi tional mean of ()() ts qzqz given the regressor is replaced by the fitted value of () qz i.e., 12 00 (())(()),(,,...)k titit sisisitisEqeyxqeyxxxqzzz where 123 (,,,...)kqzzzz is the fitted value of () qz by applying the regression o f () qz on ()ipx given H However, because too many observations are discarded in our designs, the weighted matrix () WEqqx is more likely to be singular, especially when sample size is relatively small to the number of moment functions in the conditional moment restrictions. When it happens, we cannot take the inverse of the weighted matrix, which in turn leads to unreliable estimates. Table 2 1 shows the fraction of censored observations and the fraction of discarded observations. Clearly, in our designs, about 75% of observations have censored dependent variables. In design 1, when the sample size is 200, about 63% of observ ations are discarded, only 37% of observations are used to determine the model parameters, i.e. only 75 of 200 observations are used. In design 2, about 68% of observations are not used to determine the model parameters. In design 3, fewer observations (ar ound 58%) are discarded when we increase the noise in the error terms. PAGE 64 64 I compare two estimators for 0 : the Honor estimator H and the conti nuously updating GMM estimator update For the co ntinuously updating GMM estimates, the Honor estimator is used as the starting value for the simulation to make the comparison easier. The simulation studies for EL estimator and 2 -step GMM are also implemented. For EL estimators, the computations, howeve r, are very difficult and time consuming. Considering the asymptotic equivalence of update and ,1EL and that Newey and Smith (2004) already provide the results for the smooth moment functions11, I focus on update in the paper. For 2 step GMM, I use the Charlier, Melenberg, and Soest (2000) method set H as the star ting point and go one Newton Raphson step towards GMM12. 2 -step GMM results are not presented in the paper due to its poor performance. The results for the esti mator are presented in Table 2 2 to 2713. Table 2 2 and 23 shows th e results from design 1, Table 2 4 and 2 5 shows the resu lts from design 2, and Table 2 6 and 2 7 shows the results from design 3. Tables have the contents as follows: the true value of the parameters (True), estimated mean bias (Mean_Bias), standard deviation (Std.), median bias (Median_Bias), root mean squared error (RMSE)14, inter -quartile of the parameters (the difference between 75th and 25th percentiles of the sample ) (IQR), mean absolute deviation of the parameters (MAD), and the difference between the maximum and the minimum va lue of the parameter (Range). 11 Our moment functions are not continuously differentiable; therefore, the results should contribute to t he literature by either confirming or refuting the existing results. However, due to the computational difficulty, we couldn't get the results by now; the study should be done in the future. 12 This yields an estimator () ()/H HH g g which is as ymptotically equivalent to 2 step GMM. 13 All the results reported in this paper were performed by Matlab 7.0.4 14 A RMSE value closer to 0 indicates a better fit. This statistic is also known as the fit standard error and the standard error of the regressio n. RMSE measures the average mismatch between each data point and the model. PAGE 65 65 In all three designs with different sample size, when the co ntinuously updating GMM estimate uses the moment functions ( 2 6) (where 11 k and 22 k ), which are exactly the same moment functions used by the Honor estimate, the continuously updating GMM estimator is identical to the Honor estimator. In design 1, for a small sample size of 200, the performance of the updating GMM is worse than the Honor estimator in most cases of that more moment functions are used. Only in case of 5 moment functions are used (i.e. 11 k and 25 k ), the updating GMM performs slightly better. When a large sample size of 500 is applied, in terms of RMSE, Std., IQR, MAD and Range, the performance of the updating GMM is slightly better if more but not too many mome nt functions are used. The mean bias and median bias, however, are still greater than that of the Honor estimator. If too many number of moment functions, relative to the sample size, are used in estimates, for instance (2 15), (3 9 ) or (3 15 ), the Hon or estimator outperforms the continuously updating GMM. If the sample size is further increased to 1000, the Monte Carlo evidence suggests that the updating GMM performs better except the case of 13 k and 215 k S imilar to design 1, when the large sample size of 500 and 1000 is applied in design 2, the RMSE, Std., IQR, and MAD are smaller than those of the Honor estimator in most cases. And when the sample size is set to be 200, the updating GMM performs worse tha n the Honor estimator if more moment functions are used. In design 3, the performance of the updating GMM is improved. For a sample size of 200, the updating GMM performs better than the Honor estimator in both the case of ( 11 k 25 k ) and ( 11 k 29 k ). And when the sample size is set to be 500 or 1000, the results of the updating GMM are better than that of the Honor estimator. PAGE 66 66 To summarize, the performance of the co ntinuously updating GMM is bad in all designs. This may be because all estimators we studied in this paper are based on the conditional moment restrictions ( 2 4) which require discarding part of the sample. As we described in section 2, the conditional mom ent functions in ( 2 4) only uses observations where the dependent variable is not censored in both periods as well as observations where the dependent variable is censored in one period but not in the other period and where no trimming occurs. Other observ ations would result in zero in the left hand side of ( 2 4) therefore, these observations make zero contribution in the estimation. In all three designs, over 60% observations are discarded by Honor's artificially censored residuals. With too few observat ions can be used in the estimation, the weigh ted matrix used in GMM estimator would be imprecise estimated and more likely to be singular. Thus the computation of the inverse of weighted matrix is difficult in this case. For the 2 -step GMM estimates, it's hard to compute the inverse of weighted matrix in ( 2 7 ) since it is close to zero given the Honor estimator, and the results are very poor. The computation difficulty and time consuming of the EL estimates would be caused by similar reason. Although the u pdating GMM estimator has been successfully computed, we suggest that the sample discarding problem lea ds to unreliable estimates. This finding is supported by the improve ment of the performance of the continuously updating GMM estimator in design 3. Compa red to design 1, the performance of the continuously updating GMM is improved. It is because fewer observations are discarded in design 3 (see Table 2 1 ), which is made possible by imposing more heteroskedasticity in the error terms. Although the continuou sly updating GMM behaves not well in all designs, there is some change between different designs. Comparing the results of design 1, 2, and 3, the performance of the continuously updating GMM seems depend on the variance imposed on the repressors and error terms. According to our settings, the sample generated in design 1 PAGE 67 67 has larger variance than the sample generated in design 2, and the sample from design 3 have more heteroskedasticity than design 1. However, the continuously updating GMM performs the best in design 3, and the worst in design 2. This may suggest that the continuously updating GMM behaves better under heteroskedasticity in the finite sample. Conclusion This paper studies finite sample performance of several estimators proposed for the panel data Tobit r egression model with fixed effect including the Honor estimator, the continuously updated best GMM estimator, and the empirical likelihood estimator. The continuously updated best GMM estimator and the empirical likelihood estimator could use more moment restrictions than the Honor estimator and consequently are more efficient than the Honor estimator in large samples. We compare the continuously updating GMM estimator, which has similar finite sample performance as the empirical likelihood estimator, and the Honor estimator via a Monte Carlo simulation study. By using moderate sample sizes, the results show that the up dating GMM estimator outperforms the Honor estimator with an appropriate number of moment functions. And with large sample sizes, e.g. N=1000, even using large numbers of moment functions, the updating GMM performs slightly better than the Honor. However the updating GMM estimator performs worse than the Honor estimator for most cases w hen the sample size is small ( e. g. N=200). Therefore, we suggest that increasing the number of moment functions used in estimation does not automatically lead to a large increase in efficiency, unless the sample size is very large relative to the number of moment functions used. This is may be because all estimators studied in this paper are based on the moment restrictions ( 2 4) which require discarding observations. And in our designs, close to seventy percent of observations are discarded. Having too many discarded observations would lead to PAGE 68 68 an imprecise weighting matrix estimate, which consequently leads to an unreliable best GMM estimator. It would therefore be intere sting to do another simulation study with fewer discarded observations by changing the data generated process. For example, we could decrease the fraction of discarded observations to be less than 15% by adding a positive constant to dependent variable. Ou r study also suggests an alternative estimation method (such as conditional m aximum likelihood estimation ) that does not rely on trimming for the panel data Tobit regression model with individual effects PAGE 69 69 Figure 2 115. Discarding observations when 00ix 15Observations in the set C, i.e. 12(,) yyC are discarded. PAGE 70 70 Figure 2 2. Discard ing observations when 00ix PAGE 71 71 Table 2 1 Censored o bservations a nd d iscarded o bservations Design N Observations with censored dependent variable 16 Discarded observations 17 Number of observations18 Fractions Number of observations Fractions 1 200 143.772 71.89% 125.605 62.8% 500 359.372 71.87% 314.282 62.86% 1000 719.762 71.98% 629.451 62.95% 2 200 152.1470 76.07% 134.4740 67.237% 500 380.9140 76.18% 337.6330 67.53% 1000 762.2740 76.23% 675.4030 67.54% 3 200 145.6350 72.82% 116.9160 58.46% 500 363.9830 72.8% 292.8280 58.57% 1000 729.106 72.91% 585.794 58.58% 16 We count the observations whose dependent variable is censored in at least one period. 17 Discarded observations are those observations whose artificially censored residuals for different periods equal Therefore, these observations are not used to determine the model parameters 18 This column gives the mean of the number of observations with censored dependent variable for 1000 replications PAGE 72 72 Table 2 2 Monte Carlo s tudy for the Honor a nd the updating GMM e stimator beta1 i n design 1 N (k1, k2) True Mean_Bias Std. Median_Bias RMSE IQR MAD Range 200 Honor N/A 1 0.01566 0.16224 0.00857 0.16316 0.21037 0.12651 1.1147 0 Updating GMM (1,2) 1 0.01566 0.16224 0.00857 0.16316 0.21037 0.12651 1.1147 0 (1,5) 1 0.01262 0.15771 0.00005 0.15837 0.20261 0.1224 0 1.3487 0 (1,9) 1 0.02031 0.14475 0.01115 0.14632 0.18757 0.11344 0.9973 0 (1,15) 1 0.02582 0.16644 0.0107 0 0.1686 0 0.22817 0.13286 1.1111 0 (2,5) 1 0.03742 0.18202 0.0083 0 0.18602 0.21322 0.13604 1.7612 0 (2,9) 1 0.03557 0.18126 0.01845 0.18491 0.22419 0.13881 1.6181 0 (2,15) 1 0.03606 0.19232 0.0157 0 0.19587 0.24563 0.15039 1.3758 0 (3,5) 1 0.03463 0.1794 0 0.018 00 0.1829 0 0.21909 0.13893 1.3585 0 (3,9) 1 0.04032 0.18746 0.02485 0.19195 0.23382 0.14546 1.4859 0 (3,15) 1 0.04010 0.20686 0.02725 0.16095 0.26412 0.16212 1.5226 0 500 Honor N/A 1 0.00619 0.09712 0.0054 0 0.09742 0.13753 0.07760 0.6196 0 Updating GMM (1,2) 1 0.00619 0.09712 0.0054 0 0.09742 0.13753 0.07760 0.6196 0 (1,5) 1 0.00572 0.09359 0.0008 0 0.09387 0.12139 0.07388 0.6479 0 (1,9) 1 0.00204 0.07884 0.001 00 0.07895 0.10525 0.06316 0.49347 (1,15) 1 0.0048 0 0.08834 0.00240 0.08857 0.12056 0.07095 0.58115 (2,5) 1 0.00523 0.08807 0.00042 0.08832 0.11061 0.06768 0.80576 (2,9) 1 0.00607 0.08563 0.0028 0 0.08594 0.11049 0.06716 0.55816 (2,15) 1 0.01005 0.10374 0.0028 0 0.10434 0.13394 0.08222 0.83296 (3,5) 1 0.00840 0.09043 0.00125 0.09091 0.12087 0.07176 0.60708 (3,9) 1 0.01004 0.09853 0.00255 0.09915 0.12574 0.07792 0.71632 (3,15) 1 0.01454 0.12182 0.0052 0 0.1229 0 0.16693 0.09854 0.74848 1000 Honor N/A 1 0.00293 0.06826 0.00031 0.06839 0.08389 0.05313 0.52887 Updating GMM (1,2) 1 0.00293 0.06826 0.00031 0.06839 0.08389 0.05313 0.52887 (1,5) 1 0.00417 0.06733 0.00131 0.06756 0.08629 0.05262 0.51924 (1,9) 1 0.00418 0.05966 0.00008 0.05988 0.07565 0.04669 0.40893 (1,15) 1 0.00515 0.06173 0.00081 0.06201 0.07666 0.04792 0.39703 (2,5) 1 0.00483 0.06062 0.00095 0.06088 0.07663 0.04728 0.48997 (2,9) 1 0.00489 0.06011 0.00020 0.06037 0.07478 0.04673 0.4784 0 (2,15) 1 0.00625 0.06451 0.00350 0.06488 0.08605 0.05095 0.48489 (3,5) 1 0.00524 0.06320 0.00036 0.06348 0.07989 0.04932 0.45097 (3,9) 1 0.00549 0.06315 0.00028 0.06346 0.08093 0.04908 0.50132 (3,15) 1 0.00702 0.07193 0.0068 0 0.07235 0.09506 0.05693 0.44601 PAGE 73 73 Table 2 3 Monte Carlo s tudy for the Honor and t he u pdating GMM e stimator beta2 in d esign 1 N (k1, k2) True Mean_Bias Std. Median_Bias RMSE IQR MAD Range 200 Honor N/A 1 0.01281 0.16016 0.00478 0.16083 0.21009 0.12564 1.127 00 Updating GMM (1,2) 1 0.01281 0.16016 0.00478 0.16083 0.21009 0.12564 1.127 00 (1,5) 1 0.01301 0.15963 0.0002 0 0.16032 0.21094 0.1246 0 1.0903 0 (1,9) 1 0.02890 0.18571 0.0186 0 0.18814 0.21452 0.14108 1.3314 0 (1,15) 1 0.03245 0.17156 0.0194 0 0.17478 0.21811 0.1339 0 1.2708 0 (2,5) 1 0.02555 0.17029 0.0109 0 0.17237 0.20223 0.12703 1.6627 0 (2,9) 1 0.03646 0.20241 0.0144 0 0.20588 0.24845 0.15248 1.9586 0 (2,15) 1 0.03226 0.19938 0.0129 0 0.20218 0.26614 0.15787 1.2474 0 (3,5) 1 0.03316 0.17794 0.01305 0.18119 0.22429 0.13742 1.2358 0 (3,9) 1 0.03505 0.20773 0.01545 0.21088 0.25558 0.15995 1.7439 0 (3,15) 1 0.04091 0.20219 0.01945 0.2065 0 0.25625 0.15513 1.6292 0 500 Honor N/A 1 0.01066 0.10032 0.0045 0 0.10098 0.12986 0.07873 0.68958 Updating GMM (1,2) 1 0.01066 0.10032 0.0045 0 0.10098 0.12986 0.07873 0.68958 (1,5) 1 0.00996 0.09609 0.00755 0.09670 0.1232 0 0.07501 0.70364 (1,9) 1 0.00762 0.10078 0.00131 0.10117 0.12197 0.07830 0.74675 (1,15) 1 0.00809 0.09345 0.00155 0.09389 0.12024 0.07274 0.82581 (2,5) 1 0.00785 0.08889 0.00225 0.08933 0.108 00 0.06839 0.74863 (2,9) 1 0.00778 0.09650 0.00062 0.09691 0.11967 0.07491 0.73062 (2,15) 1 0.00913 0.10556 0.0066 0 0.10607 0.13218 0.08245 0.91925 (3,5) 1 0.00845 0.09332 0.00187 0.09380 0.12012 0.07241 0.66668 (3,9) 1 0.01271 0.11207 0.00555 0.11291 0.13642 0.08562 0.83672 (3,15) 1 0.01901 0.12444 0.00775 0.12581 0.15312 0.09654 0.9263 0 1000 Honor N/A 1 0.00071 0.06708 0.00036 0.06715 0.09011 0.05317 0.42117 Updating GMM (1,2) 1 0.00071 0.06708 0.00036 0.06715 0.09011 0.05317 0.42117 (1,5) 1 0.00071 0.06708 0.00036 0.06715 0.09011 0.05317 0.42117 (1,9) 1 0.00063 0.06502 0.00028 0.06508 0.08544 0.05118 0.41175 (1,15) 1 0.00076 0.06733 0.00033 0.06740 0.09175 0.05384 0.39473 (2,5) 1 0.00060 0.05960 0.00046 0.05966 0.07730 0.04681 0.42425 (2,9) 1 0.00205 0.05775 0.00125 0.05785 0.07275 0.04504 0.42096 (2,15) 1 0.00233 0.06201 0.0005 0.06211 0.07624 0.04833 0.42557 (3,5) 1 0.00089 0.06202 0.00354 0.06209 0.07726 0.04847 0.39849 (3,9) 1 0.00117 0.05831 0.00024 0.05838 0.07763 0.04594 0.41712 (3,15) 1 0.00145 0.0653 0 0.00058 0.06538 0.08471 0.05175 0.42924 PAGE 74 74 Table 2 4 Monte Carlo s tudy for the Honor and the u pdating GMM e stimator beta1 in d esign 2 N (k1, k2) True Mean_Bias Std. Median_Bias RMSE IQR MAD Range 200 Honor N/A 1 0.02998 0.18898 0.01907 0.19134 0.23054 0.1424 0 1.4496 0 Updating GMM (1,2) 1 0.02998 0.18898 0.01907 0.19134 0.23054 0.1424 0 1.4496 0 (1,5) 1 0.03065 0.20247 0.00593 0.20478 0.21275 0.14067 2.8345 0 (1,9) 1 0.02984 0.1706 0 0.00522 0.1732 0 0.19748 0.1297 0 1.2848 0 (1,15) 1 0.04104 0.20058 0.02040 0.20474 0.2445 0 0.15367 1.8944 0 (2,5) 1 0.05342 0.21362 0.01393 0.22021 0.22016 0.154 00 2.1135 0 (2,9) 1 0.05576 0.20982 0.02299 0.21711 0.2419 0 0.15726 1.7693 0 (3,5) 1 0.05539 0.2186 0 0.01549 0.22552 0.23207 0.15777 2.0548 0 (3,9) 1 0.06043 0.2322 0 0.02142 0.25422 0.25203 0.15812 2.2367 0 500 Honor N/A 1 0.01079 0.11218 0.00130 0.1127 0 0.14488 0.08860 0.7004 0 Updating GMM (1,2) 1 0.01079 0.11218 0.00130 0.1127 0 0.14488 0.08860 0.7004 0 (1,5) 1 0.01017 0.10747 0.00238 0.10795 0.1362 0 0.08321 1.0072 0 (1,9) 1 0.01006 0.11063 0.0020 0 0.11109 0.13209 0.08458 1.079 0 0 (1,15) 1 0.00825 0.10226 0.0032 0 0.1026 0 0.12741 0.07871 0.87156 (2,5) 1 0.01287 0.09629 0.00177 0.09715 0.10938 0.07210 0.79344 (2,9) 1 0.01613 0.10389 0.00652 0.10514 0.1277 0 0.07846 0.95202 (3,5) 1 0.02087 0.10551 0.00759 0.10756 0.11342 0.07827 0.74586 (3,9) 1 0.02393 0.11021 0.00802 0.11274 0.12912 0.07992 0.98921 1000 Honor N/A 1 0.00486 0.07223 0.00102 0.09773 0.05681 0.50255 0.0724 0 Updating GMM (1,2) 1 0.00486 0.07223 0.00102 0.09773 0.05681 0.50255 0.0724 0 (1,5) 1 0.00579 0.07042 0.0012 8 0.08913 0.05538 0.52004 0.07066 (1,9) 1 0.00578 0.06976 0.00025 0.09176 0.05493 0.4349 0 0.07000 (1,15) 1 0.00542 0.06440 0.00113 0.08738 0.05105 0.43706 0.06463 (2,5) 1 0.00503 0.05955 0.00082 0.07714 0.04634 0.44918 0.05976 (2,9) 1 0.00618 0.05935 0.00163 0.07805 0.04671 0.41927 0.05967 (3,5) 1 0.00692 0.06207 0.00164 0.08108 0.04854 0.43335 0.06246 (3,9) 1 0.01119 0.07085 0.00560 0.08955 0.05512 0.457 00 0.07173 PAGE 75 75 Table 2 5 Monte Carlo s tudy for the Honor and the u pdating GMM e stimator beta2 in d esign 2 N (k1, k2) True Mean_Bias Std. Median_Bias RMSE IQR MAD Range 200 Honor N/A 1 0.02127 0.20196 0.00875 0.20307 0.21788 0.14604 2.4639 0 Updating GMM (1,2) 1 0.02127 0.20196 0.00875 0.20307 0.21788 0.14604 2.4639 0 (1,5) 1 0.02063 0.18644 0.00124 0.18758 0.21198 0.13921 1.4236 0 (1,9) 1 0.04671 0.30548 0.00575 0.30903 0.23216 0.16631 7.0522 0 (1,15 ) 1 0.0386 1 0.20708 0.00904 0.21065 0.23562 0.15198 2.1431 0 (2,5) 1 0.0480 2 0.2173 0 0.01179 0.22255 0.22026 0.15502 2.1741 0 (2,9) 1 0.05514 0.23158 0.01143 0.23806 0.25752 0.1687 0 2.0511 0 (3,5) 1 0.04436 0.21002 0.01561 0.21466 0.22812 0.15629 1.804 0 0 (3,9) 1 0.05631 0.24145 0.01753 0.24222 0.26811 0.17023 2.273 0 0 500 Honor N/A 1 0.01276 0.10464 0.00635 0.10541 0.13734 0.08175 0.69287 Updating GMM (1,2) 1 0.0127 6 0.10464 0.00635 0.10541 0.13734 0.08175 0.69287 (1,5) 1 0.01233 0.1063 0 0.00237 0.10702 0.13121 0.08147 0.86188 (1,9) 1 0.01426 0.11732 0.00093 0.11819 0.13958 0.08849 1.0294 0 (1,15 ) 1 0.01154 0.1006 0 0.00126 0.10126 0.12267 0.07843 0.74408 (2,5) 1 0.0107 7 0.09768 0.00064 0.09827 0.11696 0.07352 0.87717 (2,9) 1 0.01906 0.11788 0.00466 0.11941 0.13344 0.08814 0.89096 (3,5) 1 0.01737 0.10736 0.00570 0.10876 0.12249 0.07977 0.91179 (3,9) 1 0.01986 0.12191 0.00607 0.12641 0.13972 0.09021 0.94951 1000 Honor N/A 1 0.00671 0.07745 0.00018 0.10387 0.06100 0.62988 0.07773 Updating GMM (1,2) 1 0.00671 0.0774 5 0.00018 0.10387 0.06100 0.62988 0.07773 (1,5) 1 0.00621 0.07254 0.00202 0.09725 0.05751 0.53026 0.07280 (1,9) 1 0.00674 0.07412 0.00127 0.10023 0.05902 0.57015 0.07442 (1,15 ) 1 0.00386 0.06254 0.00414 0.08518 0.04955 0.44351 0.06265 (2,5) 1 0.00387 0.06086 0.00097 0.08366 0.04870 0.47413 0.06097 (2,9) 1 0.00519 0.06908 0.00162 0.08988 0.05476 0.46083 0.06927 (3,5) 1 0.0057 6 0.06415 0.00329 0.08494 0.05014 0.44782 0.06440 (3,9) 1 0.0105 7 0.07358 0.00839 0.10124 0.05 833 0.55048 0.07433 PAGE 76 76 Table 2 6 Monte Carlo s tudy for the Honor and the u pdating GMM e stimator beta1 in d esign 3 N (k1, k2) True Mean_Bias Std. Median_Bias RMSE IQR MAD Range 200 Honor N/A 1 0.02692 0.21813 0.01612 0.28028 0.16857 1.5282 0 0.21979 Updating GMM (1,2) 1 0.02692 0.21813 0.01612 0.28028 0.16857 1.5282 0 0.21979 (1,5) 1 0.02235 0.2103 0 0.00210 0.26299 0.16118 1.92 00 0 0.21149 (1,9) 1 0.023493 0.17836 0.01393 0.23908 0.14094 1.3655 0 0.1799 0 (2,5) 1 0.04660 0.22372 0.01313 0.26899 0.17152 2.0368 0 0.22853 (2,9) 1 0.04734 0.22876 0.02610 0.28051 0.17436 1.7848 0 0.23361 (3,5) 1 0.05391 0.23819 0.03137 0.2805 0 0.1806 0 2.209 0 0 0.24422 (3,9) 1 0.06095 0.25246 0.04329 0.30248 0.19326 2.0537 0 0.25972 500 Honor N/A 1 0.01802 0.15001 0.00437 0.19284 0.11802 0.98186 0.15109 Updating GMM (1,2) 1 0.01802 0.15001 0.00437 0.19284 0.11802 0.98186 0.15109 (1,5) 1 0.01660 0.14282 0.00241 0.18003 0.11046 1.3458 0 0.14378 (1,9) 1 0.01201 0.11559 0.00158 0.14894 0.08971 0.931 00 0.11622 (2,5) 1 0.02536 0.14052 0.00207 0.15747 0.10463 1.0019 0 0.14279 (2,9) 1 0.01797 0.12201 0.01073 0.15153 0.09466 0.83111 0.12333 (3,5) 1 0.02378 0.13482 0.00625 0.15827 0.10281 0.9706 0 0.1369 0 1000 Honor N/A 1 0.00332 0.09022 0.0022 0 0.11314 0.07029 0.65966 0.09028 Updating GMM (1,2) 1 0.00332 0.09022 0.0022 0 0.11314 0.07029 0.65966 0.09028 (1,5) 1 0.00560 0.08953 0.0009 3 0.11541 0.06972 0.64668 0.08970 (1,9) 1 0.00502 0.07627 0.00056 0.09701 0.05989 0.49129 0.07643 (2,5) 1 0.00672 0.08392 0.00031 0.10559 0.06512 0.62865 0.08418 (2,9) 1 0.00573 0.07817 0.00029 0.09935 0.06127 0.5877 0 0.07838 (3,5) 1 0.00651 0.08569 0.00077 0.10765 0.0667 5 0.56499 0.08593 PAGE 77 77 Table 2 7 Monte Carlo s tudy for the Honor and the u pdating GMM e stimator beta2 in d esign 3 N (k1, k2) True Mean_Bias Std. Median_Bias RMSE IQR MAD Range 200 Honor N/A 1 0.02151 0.21566 0.00341 0.27877 0.16892 1.501 00 0.21673 Updating GMM (1,2) 1 0.02151 0.21566 0.00341 0.27877 0.16892 1.501 00 0.21673 (1,5) 1 0.02129 0.21159 0.00091 0.26777 0.16552 1.5364 0 0.21266 (1,9) 1 0.03233 0.23275 0.01531 0.29232 0.1799 0 1.698 00 0.23499 (2,5) 1 0.02918 0.22665 0.01122 0.26313 0.17128 2.0183 0 0.22853 (2,9) 1 0.04647 0.26001 0.02214 0.32298 0.19748 2.5129 0 0.26414 (3,5) 1 0.04283 0.25517 0.01293 0.29934 0.19167 1.9254 0 0.25874 (3,9) 1 0.05274 0.27588 0.01126 0.33927 0.21268 2.0846 0 0.28088 500 Honor N/A 1 0.01902 0.13937 0.00823 0.17314 0.10781 0.95435 0.14066 Updating GMM (1,2) 1 0.01902 0.13937 0.00823 0.17314 0.10781 0.95435 0.14066 (1,5) 1 0.01779 0.13873 0.00533 0.16853 0.10541 1.0687 0 0.13987 (1,9) 1 0.01636 0.14855 0.00062 0.17259 0.11208 1.3588 0 0.14945 (2,5) 1 0.01985 0.13646 0.00589 0.16029 0.1019 0 1.235 00 0.1379 0 (2,9) 1 0.02239 0.15455 0.00510 0.18469 0.11665 1.0798 0 0.15617 (3,5) 1 0.02006 0.13821 0.00502 0.15623 0.10403 0.96338 0.13966 1000 Honor N/A 1 0.00173 0.08862 0.00037 0.11819 0.07031 0.57762 0.08864 Updating GMM (1,2) 1 0.00173 0.08862 0.00037 0.11819 0.07031 0.57762 0.08864 (1,5) 1 0.00178 0.08542 0.00167 0.11444 0.06727 0.55091 0.08544 (1,9) 1 0.00183 0.08777 0.0004 0 0.11928 0.06987 0.5156 0 0.08779 (2,5) 1 0.00379 0.07774 0.00093 0.09637 0.06077 0.59907 0.07783 (2,9) 1 0.00315 0.08269 0.00217 0.10576 0.06495 0.55477 0.08275 (3,5) 1 0.00227 0.07854 0.00136 0.10428 0.06150 0.58639 0.07858 PAGE 78 7 8 Table 2 8 A 200 O bservations Sample 1y 2y E r dif19 E r120 E r221 1Hx 2Hx 0.7284 1.3859 0.7284 1.2296 0.5012 0.5012 3.4312 0 0.6889 0.6888 0.7528 1.4416 0.1409 0.7528 0 0 0 1.9336 1.9336 0.3042 1.9336 0 2.6306 1.9842 0.2096 2.1938 0.2096 0.4369 0.6219 4.4099 0.1107 0.6867 0.7974 1.3086 5.2073 2.0017 0 0 1.7496 1.7496 3.278 1.7496 0 0 0 2.164 2.164 0.3354 2.164 0 0.4988 0 2.6909 2.6909 2.6909 2.6105 1.7654 2.4429 1.3214 0.2747 1.0466 1.4907 3.4895 0 0 0 2.1041 2.1041 1.064 2.1041 0 0 0 0.6630 0 .6630 0.0808 0.6630 0 0 0 2.1257 2.1257 1.0786 2.1257 0 0 0 2.6812 2.6812 2.6812 2.5731 9.9717 5.3846 2.8503 3.9703 1.12 6.0014 4.2646 0 0 0 1.7309 1.7309 0.0923 1.7309 5.3358 4.2358 2.1433 3.383 1.2397 1.9529 2.9962 2.5881 4.4284 1.0858 2.6011 3.6869 0.0130 0.7414 6.7975 0 0 1.3346 1.3346 7.6071 1.3346 0 0 0 1.9118 1.9118 1.9118 0.2117 0 0 0 1.5491 1.5491 0.1226 1.5491 7.3974 0.3186 3.1876 3.2382 0.0506 4.1591 0.2680 0 0 0 3.1937 3.1937 2.8143 3.1937 0 3.5761 0 1.4059 1.4059 1.4059 2.3176 0.3108 2.8436 0.3108 1.1429 0.8322 0.83215 2.3319 0 0 0 0.608 0.608 0.5082 0.608 4.9005 0.6691 0.1689 0.5783 0.4094 4.3222 0.2597 4.7071 2.9592 0.0669 1.0871 1.0202 3.62 1.9391 5.8177 3.2097 4.7453 4.0499 0. 6955 1.7679 3.9052 1.9682 0 1.2692 2.0434 0.7743 0.0752 0.7743 2.4877 0 0.0914 1.4182 1.3268 1.0695 1.3268 4.0126 2.2597 0.6668 2.2837 2.9505 1.7289 0.6909 0 0 0 1.0257 1.0257 0.2397 1.0257 0 0 0 0.9603 0.9603 0.6988 0.9603 1.561 0.0145 0.6443 0.2890 0.9333 1.8499 0.9478 0 0 0 1.5998 1.5998 1.5998 0.83104 3.1159 8.9506 1.055 2.1875 3.2424 0.9284 5.7082 2.3776 0.9126 0.9126 0.1957 1.1083 2.3481 0.1957 0 0.2306 0.2306 1.7671 1.9977 0.76913 1.7671 19 This is the difference between the artificially censored residuals of period t and s. 20 This column gives the artificially censored residuals for period 1 given the Honor estimator. 21 This column gives the art ificially censored residuals for period 2 given the Honor estimator PAGE 79 79 CHAPTER 3 MAXIMUM LIKELIHOOD E STIMATION OF PANEL DATA TOBIT MODEL Introduction The c ensored regression model is an important and interesting econometric model in econometric applications. A natural mod el for analyzing panel data containing censored dependent variable is the panel data Tobit model with individual effects We focus on the following Tobit model: 0 0, 1 if 1 ,1,2,....,;1,2,...., 0, if 0itiitit itiitit itit it ityx dx yd y iNtT d (3 1) where i denotes the individual, t denotes time; ity denotes the latent dependent variable; ity denotes the observed dependent variable; itx denotes the k dimensional vector of time variant explanatory variables; i denotes the unobserved individual speci fic effect s ; 0 denotes the true value of the unknown parameter vector to be estimated; it denotes the error term, and itd denotes a indicator function. Estimation of panel data Tobit model (3 1) can be difficult, depending on the assumpt ions imposed on the error term and the individual effect. For assumption about the error terms distribution i n empirical work we usually assume that it follow s the normal d istribution 2 0(0,) N 1; and for different time periods, conditional on the explanatory variables and individual effects, we assume that the error terms are identically distributed For the individual effect, when it i s allowed to correlate with explanatory variables arbitrarily, the estimation of 1 For simplicity, we assume that 2 0 is fixed. PAGE 80 80 model (3 1) is nontrivial and difficult since it enters the model nonlinearly and thus simple time differencing cannot remove it Several estimators have been proposed for Tobit model ( 3 1). Chamberlain (1984) uses a parametric specified model to derive his estimator. His estimator could provide some interesting quantities such as marginal effect of covariates on the observed censored variable. However when the parametric mode l is misspecified Chamberlain s estimator will, in gene ral, be asymptotically biased. To ove rcome this problem, Honor (1992) offers a semi parametric model whi ch avoids assumption of the distribution. Under some conditions, Honor's estimator is consisten t and asymptotically normally distributed The drawback of his estimator is that the estimator is not efficient since it does not use all moment restrictions. To increase the efficiency of the estimates, the two -step GMM, t he continuously updating GMM, and the empirical likelihood estimator (ELE) exploit other moment restrictions. Asymptotically, the latter three estimators are equivalent and are mor e efficient, at least not worse than Honor's estimator. But, all three estimators are hard to compute due to the non-smooth problem and require discarding observations Th e latter three approaches rely on artificial censoring to restore the zero correlation between the explanatory variables and the time -differenced artificially censored residuals. However, when most individual observations are discarded as we discussed in chapter 2, these estimators have poor finite sample performance. Thus, it is imperative to find an alternative estimator that do es not rely on trimming but is still consistent and asymptotically efficient. In this paper, w e consider the conditional maximum likelihood estimation (MLE) for model ( 3 1) given the normality assumption on the error term. N otice that we dont use conditional moment restrictions (2 4) in lo g likelihood function which means that conditional PAGE 81 81 MLE does not rely on tri mming. However, there is risk in implement ing the MLE method due to the presence of unobserved individual effects. Since we don't know the distribution of individual effects, the m isspecification of the parametric form of the density that defines the log likelihood function can seriously bias estimate of model parameters. To overcome the problem of unknown density, we consider the semi parametric estimation proposed by Gallant and Ny chka (1987) which can consistently approximate an unknown density function. The main objective of this art icle is to implement Gallant and Nychka s (1987) semi parametric estimation to consistently approximate the unknown density for individual effects and then propose a consistent MLE estimator for unknown parameters in model (3.1) and unknown density of individual effect s while also deriving the asymptotic distribution for model parameters in (3.1) by using Shen 's (1997) technique for sieve estimation The res t of the paper is organized as follows. Section 2 formally introduces the settings for maximum likelihood estimato r. Section 3 shows the consistency of the ML E estimator and computes its conve rgence rate. Section 4 derives the n a symptotically normal ity of the ML E estimator for unknown parameters in the model (3.1) Section 5 provides a consistent covariance estimator for the model parameters estimator and Section 6 concludes the paper All technical proof s are presented in the Appendix. MLE Estimator For simplicity, we will drop the subscript i and only consider two periods (i.e. T=2) in this study Denote 12, xxx and 12, y yy S uppose that 1,2,..., yxin is a sample of observations and is drawn from the true density 0() fyx with support of YX, where Y is a subset of ydR and X is a compact subset of xdR Let z denote all observed time -invariant PAGE 82 82 explanatory variables including x S uppose that the error term t is independent of z and conditional on i the error terms 1 and 2 are independent. Also assume that conditional on z has an unknown de nsity function 0( z) h Let and denote the standard normal probability distribution and density function respectively. F or each individual i it is easy to compute that the conditional joint density of y given (,) x is: 2 00 1 0001 (, )(1)(1())(( ))t tt tt txyx fyx d d Hence, we can write the conditional density function of y given z a s : 2 00 00 1 0001 (,)(1)(1())(( ))()t tt tt txyx fyz d d hzd (3 2) where 0 denotes the unknown coefficients 000,, h 2. Then the log likelihood function is 2 00 00 1 0001 log(,)log(1)(1())(( ))()t tt tt txyx fyz d d hzd (3 3 ) T h erefore the parameters of interest 0000(,,) h contains a vector of finite dimensional unknown parameter s 00(,) where 00,dRR and an infinite dimensional parameters of density 0h The parameter space is the set of admissible density functions and defin ed by Gallant and Nychka (1987). For simplicity, t hrough the paper, denote (,,) h and Under the assumption that Tobit model ( 3 1) identifies 0 we propose to estimate 0 b y conditional MLE. Heuristically, if the function form () hz the conditional density of given z PAGE 83 83 w ere known, then the function form of the conditional density of y given z (,) fyz would be known. The true value 0 solves the following constrained problem: maxlog(,) subject to (), ()0. Efyz hzdhz (3 4 ) The true value of 0 could then be estimated by maximizing the sample analog of ( 3 4 ). However, s uch an optimization procedure (3 4) is difficult to implement because t he specified form of () hz and therefore (,) fyz is in fact unknown. Moreover, the density function () hz is infinitly dimen s ional and it is impossible to be estimated from fin ite data points when the space is too large. Often, optimization over a large parameter space leads to inconsistency or roughness. To overcome the problem we use sieve approximation for the unknown density function () hz introduced by Gallant and Nychka (1987). Their idea is to use a smoothness assumption to obtain an analytically tractable series representation for the unknown density () hz by replacing with a sieve space k More specifically, the optimization is carried out within a sieve space k w hich is a computable and finite -dimensional compact paramet er space that is dense in the original space as k increases. Let k be a sequence of approximating spaces to (not necessarily a subset of ), denote d as a sieve, in the sense that for any h there exists kkh such that 2()()0 as khzhzk 2 It is important to see that (3 3) does not depend on the individual effect has been integrated out. PAGE 84 84 Following Gallant and Nychka (1987), the common choice of () hz is a series expansion 22()()(,)kkhzPz Let (see Ai and Chen for an () hz is restricted to 22 2 0 2()()(,)(),0,1,..., and ()=()C kkk k kkhzPz Zk H hzhzd (3 5 ) W e propose to take 2 0121 ()exp(( ...)) 2k kkP therefore 22 012()()exp( ...)k kkhz Obviously 0,...,k are functions of z i.e. 00(),...,()kkzz for 1,2,... k Let 1()kpz denote a vector of known basis functions of degree 1k that can approximate any square integrable function of z arbitrarily well, a nd j denote a vector of coefficients. Thus, for each arbitrarily fixed 1,2,..., jk ()jz can be approximated by 1()kjpz for some vector of coefficients j as 1k i.e. 1 1 11 1 1. ()(),...,() () .j k j k sjs s jkzpzpz pz (3 6 ) The above proposed specification form for ()khz has considerable computational advantages. First, ()khz is guaranteed to be nonnegative everywhere in our settings Next, we need to consider the density restriction ()1khzd To ensure that ()1khzd we solve 22 0 12ln()exp( ...)k kd PAGE 85 85 Substituting 0 back into ()khz we obtain 22 12 22 12()exp( ...) () ()exp( ...)k k k k khz d (3 7 ) where j takes the form (3 6 ). Clearly, the proposed form ( 3 7 ) of ()khz s atisfies the density restriction s i n the maximum problem (3 4 ) Now, t he semi parametric approximation to the density function () fyx with () hz replaced by a sieve estimator ()khz is: 2 11 (,)(1)(1())(())()t tt kttk txyx fyz d d hzd Denote kk which is a finite dimensional approximation to T o summarize, the conditional ML E estimator of 0 maximizes the sample analog of the parametric version of ( 3 4 ) with ()khz restricted to the sieve space k : 2 1 11 argmaxln(1)(1())(())()kn t tt ttk i txyx d d hzd (3 8 ) The advantage of the conditional MLE procedure ( 3 8 ) is that it is natural and easy to implement. Once h is replaced by kkh the estimation problem effectively becomes a parametric one; hence commonly used econometric software packages can be used to compute the estimator In addition, it is important to see t hat (3 8) does not depend on conditional moment restrict ions, so we dont need to discard observations that are not satisfied by the conditional moment restrictions and worry about the trimming problem3. 3 As we discussed in Chapter 2, when most of individual observations cannot satisfy conditional moment restrictions (2 4) and we have to discard them in estimation, the estimation will a give bad performanc e estimator. PAGE 86 86 In the following sections, we show the consistency o f and kh and derive the asymptotic distribution for Our main results are A 1 0 1 4 0 2 1 2 4 0 00 2 ()(0,) () (()())()()p kk pN NV On hhhhdOn Consistency We begin by introducing additional notation to aid the exposition. Denote (,,)k sN as the minimal number of radius covering balls of k under the metric s it measures the size of k .The estimator we consider is the one k from (3 8 ). Le t (,)ln(,)klyfyz and define 11 ()(,)n n iLly n ()(,) LEly Let s denote the pseudo metric on given by 0000 2 sEEhh (3 9 ) where E denotes the Euclidean n orm, and 2 00 2[] hhhhd In this section, we first present sufficient conditions and apply the result s of Gallant and Nychka (1987) to obtain consistency of the ML E estimator for 0 under the metric s We then establish the convergence rate under a we aker metric The convergence rate result will be used in Section 4 to establish the asymptotic normality for PAGE 87 87 Assumption 1: (a) For each t conditional on ,, i=1,2,...,nitityx are i.i.d, and satisfy the Tobit model ( 3 -1); (b) ()p tEy exists and finite for some 2 p ; Assumption 2: (a) X is compact with nonempt y interior ; (b) the support of z is bounded; Assumption 3: (a) The support of is a subset of the finite interval 12, ; (b) 0() hz is bounded and bounded away from zero; Assumption 4 : the error term is it independent of z and for each t it are i.i.d. error, and follows the normal distribution 2 0(0,) N ; Assumption 5 : 0 dR is compact with nonempty interior ; Assumption 6: T here exists finite constants 1 and 2 such that 1020 ; Assumpti on 7 : 120()1tx holds for all x and 012[,] ; Assumption 8 : 0000(,,) h is the only solution of for maximum problem (3 -4 ). Assumption 1 rules out time series observations. Assumption 1(b) is needed for consistency proof. This condition is common and requires the existence of moments. Assumption 2 is a typical condition imposed for sieve estimation, and can always be satisfied by discarding large values of the reg ressors Assumption 3 (a) restricts the suppo rt of individual effects and Assumption 3 (b) bounds the unknown density 0() hz away from zero. Assumption 4 is commonly imposed in empirical work for the error term. Assumption 5 is commonly imposed in the literature for compactness of parameters This condition is needed to prove consistency. Assumption 6 requires that is bounded and bo unded away from zero Assumpt ion 7 basically PAGE 88 88 requires normal distribution ()tx is bounded and is needed to prove consistency as well as derive the asymptotical normality. And Assumption 8 is an identification condition4 To prove consistency first we need to verify the compact ness which is commonly imposed in the nonparametric and semiparametric econometrics literature. Clearly, f ollowing assumptions 5 and 6 is compact in the topology generated by the norm EE Then t he compactness is satisfied when the infinite -dimensional parameter space consist s of bounded and smooth functions. Recall that the definition of follows by Gallant and Nychka (1987), by applying the result s of their Theorem 1; the closure of is compact under the norm 2h .Thus, t he closure of is compact under the norm s Next, t he denseness condition requires that the sieve spaces k approximate the parameter space Obviously, in our settings, 1 kk Applying Theorem 2 of Gal lant and Nychka(1987), we obtain that 1 k k is dense in the closure of with respect to 2h Clearly, by the definition of kk 1 k k is dense in the closure of with respect to s Re call that in Tobit model ( 3 1), variables are generated according to nonlinear regression. To show that the convergence is uniform we need s tochastic domina n t and Lipschitz condition Rewrite the log likelihood fun ction (,) lyz as (,)ln(,)ln(,,)()kklyfyzgyzhzd where 4 Gallant and Nychka(1987) gave the proof of identification condition for the example of the sample selection problem in section 4. Their technique could be used to prove the identification condition in our settings. PAGE 89 89 00 01 10 11(,,)(,,)(,,)(,,)(,,) gyzgyzgyzgyzgyz 12 00 12(,,)(1)(1)(1())(1()) xx gyzdd 122 01 121 (,,)(1)(1())(( )) xyx gyzdd 11 2 10 121 (,,)(1)(())(1()) yxx gyzdd 1122 11 1211 (,,)(())(( )) yxyx gyzdd Following by A ssumption 1 8 in the appendix we show that (,) ly is stochastic ally domi n ated and Lipschitz continuous From Newey (1991), with envelope condition, Lipschitz condition and compactness condition, we have sup ()() (1)npLLo over k and () L is continuous in The denseness condition and the continuity of () L ensure that ()()nLL uniformly over A pplying Theorem 0 of Gallant and Nychka (1987), we have : Theorem 1 Suppose that are obtained by maximizing (38 ), and Assumption1 8 hold, we have 0 (1)p so Theorem 1 is proved in the appendix. Theorem 1 provide s a consistency result under the pseudo metric s but it is not enough to establish the asymptotic normality of I t is wellknown that to derive the asymptotic distr ibution, we typically need the convergence rate of the estimator to be faster than 14n T he convergence rate tell s us the curvature of the criterion function. PAGE 90 90 We now introduce another (weaker) metric and compute the convergence rate of the estimator under this metric. L et () E and var() denote the expectation a nd variance To find the convergence rate n we need to show that 12()pno for any 12,k Suppose that the parameter space k is connected in the sense that for any two points 12,k there exists a continuous path 00()():[0,1]k Thus, 0(0)(1) Clearly, in our settings, 00 (,()) ly is continuously differentiable at 0 Denote the directional derivative of (,) ly at 0 by: 00 00 0(,()) ()[] ly l Notice that 00 0 000 000(,()) ln(,) ln(,) ln(,) ()()[]kkk kly fyz fyz fyz hh h Now, a pplying a Taylor expansion, for all k and all z we could write 0000(,)(,)()[](,) lylyl r (3 1 0 ) where 0(,) r is the remainder term Hence, the average Kullback Leibler information is : 2 00 00 00 2(,()) 1 (,)(,)(,) (,)2 Elz KElzElz R since 00()[]0 El Notice that the equality 2 2 00 00 00 2(,()) (,()) 0 ly Ely E PAGE 91 91 holds for any For any 12,k define the metric as: 2 2 000 2 000 000()[] ln(,) ln(,) ln(,) ()()[]kkk kEl fyz fyz fyz E hh h (3 11) and the fact that 00(,)(,)()slylycz where 00 2 s imply that 00 sc Notice that the remainder 0(,) r is in the order of 3 2 Thus, in the neighborhood of 0 2 0 is equivalent to t he avera ge Kullback Leibler information 0(,)(,) Elyly We now present additional conditions for computing the convergence rates under the metric : Assumption 9: 11 ln(,,) lnk sN constkk ; Assumption 10: 1()v pkkon for some 01 v ; Assumption 11: for any there exists kk satisfying 1()()v kpp sokkon Notice that u nder condition 10, we have the expec ted criterion difference reduced to the Kullback -Leilber pseudo distance : 2 00(,)(,)(,)()v pKElzlzon Hence, applying Theorem 1 of Shen and Wong (1994) we obtain the following Lemma: PAGE 92 92 Lemma: Under A ssumption s 1 11, we have (1)3 0(max(,))vv ponn The lemma is proved in the Appendix. Asymptotic Distribution Having computed the conv ergence rate of the proposed MLE estimator under we now derive the asymptotic distribution of (,) e For simplicity, we assume 2 d We f irst introduce some notations. Denote () fe for any fixed and nonzero 3R To study the asymptotic distribution, we discuss linear approximations of the criterion difference by the corresponding derivatives and the degree of smoothness of function () f Let V denote the closure of t he space spanned by 0 under the metric Suppose that induces an inner product on V then for any 12, vvV V is a Hilbert space with the inner product: 00 12 1 2(,)(,) [][] dlzdlz vvEvv dd (3 1 2 ) where 00 0(,)(,) [] dlzdlzv v dd By the results in Van der Vaart (1991) and Shen (1997), () fe must be bounded (i.e. 00 0 0()() supVff ) so that it could be estimated at the n rate. Thus, we need to show that () fe is bounded. In addition, by the Riesz representation theorem, there exists vV such that, for any we have PAGE 93 93 00000()()()[] fff eev (3 13) Hence, the asymptotic distribution of 0 is the same as the asymptotic distribution of 0 v By definition, 02 0 0 2 0 00 2 0 1()() sup ()() sup sup ()()V wwff v eeee EDzDz (3 14) where 00(,)(,) () []wdlydly Dz w dedh I t suffices to find w which minimize ()()wwEDzDz Denote 0Wh and VW F or each component ,1,2,3jej let jwW denote the solution to 2 00(,)(,) min [],j jdlydly w dedh (3 1 5 ) Define 123(,,) wwww 0000 123(,)(,)(,)(,) []([],[],[]) dlydlydlydly wwww dh dhdhdh we have 00(,)(,) () []wdlydly Dz w dedh (3 16) Hence, 1 2()()wwvEDzDz and 2 1(1,)(,)ehvvwvvV with 1()()e wwvEDzDz hevwv PAGE 94 94 Moreover, (3 14) implies that E()()wwDzDz has to be finite positive -definite in order for () f to be bounded. Recall that 0 0 00 000 000(,) (,) (,,) (,,) g g hd hd ly ly gyhd gyhd and 00 0 000(,,) (,) [] (,,) gywd dly w dh gyhd Thus 00 000 0(,)(,) () [] ()(,,) (,)w kdlydly Dz w dedh g hgywd e fyx implies that 00 00 00 00 00 0()(,)()(,) ()() (,) (,) =E(()(,))(()(,))(,ww kk kgg hgyewdhgyewd ee EDzDzE fyx fyx gg hgyewdhgyewdfyx ee 2 2 0) (,)kkkk kffff wwfyx eheh V (3 17) PAGE 95 95 Recall that 0(,)0kfyx therefore we need kkkkffff ww eheh is finite positive definite. Let 12 1()()()n n ii iQqnqzEqz denote the empirical process induced by any function () q And let 12()npon Assumption 12 (a) kkkkffff ww eheh is finite positive -definite ; (b) 0int() E ; Assumption 13. There is a 0(,),k kkehkhkevvv vwv such that 14()kpvvon ; Assumption 14 1 4 and 1 4 v Assumption 12(a) is a local identification condition for 0 this condition cannot be relaxed. This condition must be satisfied for the estimated finite dimensional parameter to be n -consistent. However, it is hard to verify it in practice since we don't know the true value of the model Assumption 12(b) used to satisfy that 0 is an interior point of E Assumption 13 is needed due to the presence of unknown 0h We simply assume that the same sieve space kH approximates the space 0WHh well5. Assumption 1 4 is required to ensure that the rate of MLE estimator in our set ting converges to the t rue value 0 under the norm is at least 1 4n 5 Theorem 2 could be prov ed even if hv is approximated by any other sieve spaces, possibly different fromkH. PAGE 96 96 Theorem 2 Under Assumptions 1 12, 1 0 ()(0,) nNV where E()()wwVDzDz Theorem 2 shows that under sufficient conditions, the proposed conditional MLE estimator of the parametric component 0 is n -consistent and asymptotically normally distributed. Now we compute the variance for the proposed MLE estimator of the parametric component 0 Covariance Estimator The asymptotic distribution derived above can be used for statistical inference on the parametric component 00(,) To conduct the statistical inference, we need a consistent and easy to compute estimator of the covariance matrix. In this section, we provide one such estimator by consistently estimating 1V given in section 4. The only thing we need to estimate is w given (317). For each ,1,2,3jej we approximate jw by jw which is the solutio n t o the minimization problem (3 15). Let 123(,,) wwww then the estimator of 1V is 2 1 11() () ()() ()()()n ige ge V hgewdgehd hgewd ne e Theorem 3 Under Assumption 1 12, we have (1)pVVo Conclusion In this paper, we propose a semi parametric conditional maximum likelihood estimat or for the panel data fixed effect Tobit regression model (3 1) We present some sufficient condi tions, under which we show that the MLE esti mator of the density function for individual effect s is consistent and derive the convergence rate. We also show that the estimated finite dimensional PAGE 97 97 parameter is n consistent and asympt otically normally distributed. In addition, we provide a consistent estimator of its asymptotic covariance matrix. MLE es timation uses s mooth object ive function by integral over individual effect instead of using the condit ional moment restrictions (2 4). T herefore, conditional MLE estimation doesnt need to discard observations. T he MLE estimator proposed in this paper should be more efficient than Honor's estimator, the two -step GMM, t he continuously updating GMM, and the empi rical likelihood estimator (ELE) In addition, not only can the model parameters be consistently estimated by MLE estimation but also the distributions can be consistently estimated As we showed in S ection 3, we provide a consistent estimator for the den sity functio n of unknown individual effects Also, we p rovide a consistent estimator for the error terms variance which will tell us the distribution for the error term Moreover we can compute the estimated density function for y given x. The disadvantage of the conditional MLE estimator is that we need to know the parametric specification form of the error terms distribution The normally distributed error term, though commonly imposed in empirical analysis, however, is difficult to justify. When the normality distribution assumption is not true, in general, the MLE estimator will be biased. In addition, the computation of MLE estimation is too complicated. There are several limitations in the current study It is unclear whether more efficien cy advantages truly exist in finite samples. It would therefore be interesting to do a simulation study about the finite sample performance of the MLE estimator similar to the one we did in Chapter 2 However, due to the double integral simulation for MLE cannot be implemented in MATLAB. Second, in this study, we assume that the identification condition is satisfied. It is possible to show that the identification holds for our specified model. The proof of identification will PAGE 98 98 provide a missing piece in the literature. In the future, these two parts should be finished and added to the paper. PAGE 99 99 APPENDIX PROOF FOR CHAPTER 3 P ROOF (Theorem 1). Let c denote a generic constant that may have different values in different expressions. To show envelope condition and Lip schitz continuous we will discuss four cases. Case 1: 120,0 dd Hence, 00 00(,)ln(,)ln(,,)()klyfyzgyzhzd Notice that, because the explanatory variables and the unknown coefficients are all bou nded following by assumption 1 7 we have 200 12ln(1)ln(,)2ln(1), fyz hence, 00ln(,) fyz is bounded and has bounded der ivatives with respect to and The directional derivative with respect to the unknown density h is 12 00 12(1())(1())() ln(,) [] (1())(1())() xx hzd fyz h xx h hzd and the fact that 21 011()1tx imply that 00ln(,) fyz h is bounded by 2() chz So 00ln(,) fyz is stochastically dominated and Lipschitz continuous. Case 2 : 120,1 dd In this case, we have PAGE 100 100 01 01(,)ln(,)ln(,,)()klyfyzgyzhzd Notice that for all 12, 2 2 12 22max(,) () exp( )exp()1 22 uu u We obtain 221222 21 01 21 1max( ) 11 ( ))()(,)()kkyxyx hzdfyzhzd It follows that 01ln(,) fyz is stochastically dominated if 2()pEy exists and is finite for some 2 p S ome calculations give 01 1122 212222 2(,) 1 ()(( ))() (1())( ( ))() fyz xxyx hzd xxyxyx hzd which is bounded by 1222 122 2 111 1222 01 2 111 ( )(1())(( ))() (1) (,) (1) xxyxc xyx hzd xxyxc fyz for some constant c Thus, the derivative of (,) ly with respect to is stochastically dominated by 01 1222 2 11ln(,) (1) fyzxxyxc Similarly, we find PAGE 101 101 01 1122 2 2 12222 4(,) 1 ()(( ))() () +(1())( ( ))() fyz xxyx hzd xyxyx hzd is bounded by 22 122 01 3 11(,) (1) xyxc fyz Hence, the derivative of (,) ly with respect to is stochastically dominated by 22 01 122 3 11ln(,) (1) fyzxyxc For the directional directive, we have 122 01 011 (1())(( ))() ln(,) [] (,) xyx hzd fyz h h fyz which is bounded by 2() chz Therefore, 01ln(,) fyz is stochastically dominated and Lipschitz continuous. Case 3: 12 1,0 dd Similarly, 10 10(,)ln(,)ln(,,)()klyzfyzgyzhzd Using similar line of arguments, we can show that 111112 21 10 21 1max( ) 11 ( ))()(,)()kkyxyx hzdfyzhzd and 10 2111 2 11ln(,) (1) fyzxxyxc PAGE 102 102 Thus, 10ln(,) fyz is stochastically dominated if 1()pEy exists and is finite for some 2 p and Lipchi ze continuous holds Case 4 : 121,1 dd 11 11(,)ln(,)ln(,,)()klyzfyzgyzhzd Using similar line of arguments, we can show that 11ln(,) fyz are stochastically dominated and Lipchize continuous. Therefore, the above discussion s imply that : (a ) t here exist s a measurable function 1(,) cyz with 1[(,)]pEcyz for some 2 p such that 1(,)(,) lycyz for all and y,zk (b ) for all z and 12,k there exists a measurable function 2(,) cyz with 2 1[(,)] Ecyz such that 12212(,)(,)(,)slylycyz T heorem 1 now follows from application of Theorem 0 of G allant and Nychka (1987). Q.E.D. P ROOF ( Lemma 1 ). Given 2 00 00(,()) ly E the Condition C1 of Shen and Wong (1994) is true with 1 Furthermore, after some manipulations, we know Condition C2 of Shen and Wong (1994) holds with 1 PAGE 103 103 0 2 2 00 2 0 2 2 212 2 2sup var(,)(,) (,)(,)(,)(,) (,)(,) () 2slyly ElylyElyly Elyly Ecy A To show Condition C3 of Shen and Wong (1994), we have to calculate the metric entropy function. Let () B be the same as defined in Ossiander (1987). Notice that 2 ()0 2 2 12 2sup (,)(,) ()B sElyly Ecz C for some constant 0 C Hence, let 0(,)(,):kkFlyly for some constant 30 A and all small 0 applying assumption 7 and the inequality ln()1 for all 0 xxx we have 1 31 1 31 1 31ln(,)ln(,,) ln (1) kk sNFN Akk Akk Akk Then condition C3 holds with 0, 1 2 v rr Assume 1k the lemma now follows from application of theorem 1 of Shen and Wong (1994). Q.E.D. P ROOF (Theorem 2 ). Recall the notation ()nL a nd () L i ntroduced in Section 3. To simplify notation, denote uv kkuu Denote 0()(), ()(),k nnkLu Lu PAGE 104 104 The fact that is the solution for (3 8 ) implies that ()n is maximized at 0 and 0 is the solution for (3 4 ) implies that () is maximized at 0 H ence, we have t he following first order condition : 0 0 0 1()() 0, () 1 () 0k n k n ilu E lu n (3 18) Since converges to 0 in probability, applying the empirical process theorem we have 12 0 00 ()() ()()()kk nnplulu Q Q on (3 19) Substituting (3 18) into (3 19), it follows that, 12 0 00 1 ()() 1 ()n kk p ilulu E on n Applying a Taylor expansion around 0 for 0 ()klu E we obtain 0 2 00 0 0,020 2 0 0,020 () () (()) () (()) ()k kk s k slu E lulus EE r s lus Er s Note that the remainder term 20 () r is dominated by 2 0 i.e 2 12 200()()pr on Also, given the definition of in (3 12), we have, PAGE 105 105 0 2 0 0,020 12 0 () (()) () ,()k k s kplu E lus Er s uon Notice that follows from assumption 11, and uv kkuu we have 00,,kuv 12 00 () () ()k plu lv E E on Hence, we have, 12 0 00 1() 1 ()n p ilv v on n Since 00, v we obtain Theorem 3 by applying a standard central limit theorem. 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She received her Bachelor of Arts degree in economics and Bachelor of Scien ce degree in mathematics in June 2003 from Huazhong University of Science and Technology. S he started graduate school at the University of Florida in 2003. She got her M.A. in economics in 2005 and Ph.D. in economics in 2009 from the Depar tment of Economics, Warrington College of Business Administration. In 2009, she began employment at Shanghai University of Finance and Economics. Her research interests include econometrics, strategic management, industrial or ganization, and applied econom ic s. |